Journal of Convex Analysis Volume 3 (1996), No.1, 117–140

A Bornological Approach to Rotundity and Smoothness Applied to Approximation^{1}

John Read

Center for Experimental and Constructive Mathematics, Department of Mathematics and Statistics,

Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada. e-mail: read@cecm.sfu.ca

Received June 1, 1994 Revised manuscript received November 17, 1995

Dedicated to R. T. Rockafellar on his 60th Birthday

In this paper we are interested in examining the geometry of a bounded convex function over a Banach space via its subdifferential mapping. We will consider two concepts. The first is the single valuedness and continuity of the subdifferential mapping, and the second is the single valuedness and the continuity of the “inverse” of this mapping. The smoothness of f is important for the first concept as the convexity of f is for the second. We generalize some of the well known results on upper semi-continuity of the subdifferential mapping, and we introduce a bornological approach to convexity, which allows us to draw very nice parallels for the continuity of the inverse mapping with the coresponding concept for the well understood subdifferential mapping. The theory developed allows us to give a local Smulyan result in which the convexity at a point on the unit sphere is characterized by the uniform smoothness of the subdifferential of this point, and to give the smoothness of the primal norm at a point in terms of the convexity of the dual norm about the subdifferential of that point. As the title implies we will place special emphasis on the approximation of this convex function by a sequence of such functions and derive conditions, which ensure satisfactory approximation of the subdifferential and “inverse” mappings.

Keywords : approximation, bornology, convexity, differentiability, the subdifferential mapping and its inverse, Banach space geometry, local uniform rotundity, local Smulyan results, optimal control of vibrations

1991 Mathematics Subject Classification: 26E15, 46G05, 46A17, 46A20, 46B20, 46B25 46N10, 49J20, 49J27, 49J50, 49J52, 54C08, 58C20

1. Introduction: A Motivation

The aim of this report is to describe -in terms commonly used in Banach space geometry and convex analysis- the problem that is at the heart of approximating the norm (time) minimal distributed control of a vibrating process, which is governed by an abstract wave equation (e.g. strings, beams, membranes, plates and networks of these) in a Hilbert

1 This paper is the first chapter of the author’s dissertation submitted to the Department of Mathmatics of the Technische Hochschule Darmstadt D17. The author would like to thank Professor Hofmann of said institution for his time and effort in reading this manuscript and providing many useful comments.

ISSN 0944-6532 / $ 2.50 ◯^{c} Heldermann Verlag

118 J. Read / A bornological approach to rotundity and smoothness

space. This approach leads naturally to many questions in renorming of spaces à la [4].

One can show that the problem of finding the set of all distributed control functions (that is, the problem of determining the set of all forcing functions which bring an initial state of our system to rest within a given time T > 0) boils down to solving the linear equation

S(ω) = c,  (1.1)

where S is a surjective bounded linear operator from the Bochner space L^{2}((0, T), ℓ^{2}) onto the Hilbert space ℓ^{2}(IR^{2}). The operator S is determined by the eigenelements λ_{j}, φ_{j} for j ∈ IN, of the system being examined. For example, the vibrating string of unit length fixed at both ends has eigenvalues λ_{j} = (jπ)^{2} and eigenfunctions φ_{j}(ξ) = sin(jπξ) for ξ ∈ [0, 1] and all j ∈ IN. The right hand side of (1.1) contains information about the initial state of our problem and is an element of ℓ^{2}(IR^{2}). The solution set of (1.1) is some affine subspace of L^{2}((0, T), ℓ^{2}). We wish to find any (the unique) element of this affine subspace, which for some given p ∈ [2, ∞] minimizes the norm on L^{p}((0, T), ℓ^{2}) ⊂ L^{2}((0, T), ℓ^{2}).

We introduce the Hilbert adjoint operator of S, which we denote by S^{∗}. By definition ⟨S(ω), x⟩ = ⟨ω, S^{∗}x⟩ for all ω ∈ L^{2}((0, T), ℓ^{2}) and all x := (x_{j}^{1}, x_{j}^{2})_{j∈IN} ∈ ℓ^{2}(IR^{2}), It is

easy to see that S^{∗} is given by S^{∗} = (S_{j}^{∗})_{j∈IN} where the operator S_{j}^{∗} acts on x ∈ ℓ^{2}(IR^{2}) to give

(S_{j}^{∗}x)(t) := x_{j}^{1} sin \sqrt{λ_{j}}t + x_{j}^{2} cos \sqrt{λ_{j}}t for t ∈ [0, T] a.e..  (1.2)

After some technical details we can use (1.2) to show that any (the unique) p-norm minimal solution of (1.1) has the form (which for p = ∞ is called a bang-bang control) given by

ω(t) := ⟨c, x̃⟩∥(S^{∗}x̃)(t)∥^{q−2} (S^{∗}x̃)(t) for t ∈ [0, T] a.e.,  (1.3)

where q is the dual index for p, that is we have p^{−1} + q^{−1} = 1, and where x̃ is any (the unique) solution of the dual problem in ℓ^{2}(IR^{2}), which is

to maximize ψ_{c}(x) := ⟨c, x⟩
subject to f(x) := ∥S^{∗}x∥_{q} ≤ 1.

Clearly, S^{∗} is intimately connected with the adjoint of the restriction of S to L^{p}((0, T), ℓ^{2}), and (even for p = ∞) the range of this adjoint is contained in L^{q}((0, T), ℓ^{2}). Much of our analysis is simplified by showing that f is actually an equivalent norm on ℓ^{2}(IR^{2}). We designate ℓ^{2}(IR^{2}) with norm f by Y, whose dual space Y′ is ℓ^{2}(IR^{2}) with norm f^{∗}.

Modulo an application of the Lagrange multiplier rule, we realize our wish if we can find any (the unique) x̃ ∈ Y such that x̃ ∈ S_{1} := {x ∈ ℓ^{2}(IR^{2}) : f(x) = 1}, the unit sphere in Y, and such that there exists some λ > 0 with c/λ ∈ ∂f(x̃), where ∂f(x̃) is the subdifferential of f evaluated at x̃. Of course, the subdifferential of f at any x ∈ S_{1} is contained in S′_{1}, the unit sphere of Y′, and is exactly the set of norming functionals for x. In this way the control problem is reduced to understanding the relationship between f and ∂f.

So far so good, but our standpoint is a practical one, and even in the simplest of cases one cannot expect to have exact knowledge of the right hand side of (1.1). More often one realizes the actual initial state of the medium (that is c) in some kind of limiting process. For instance, one might consider the initial state c_{L} ∈ ℓ^{2}(IR^{2}) arrived at by projecting c onto the space spanned by the first L fundamental harmonics for each L ∈ IN. That is, one truncates c as in [9]. For each c_{L} we find λ_{L} > 0 and x̃_{L} ∈ S_{1} such that c_{L}/λ_{L} ∈ ∂f(x̃_{L}). Since the sequence of approximating control functions is defined in terms of the sequence {x̃_{L}}, the question of convergence of {x̃_{L}} to x̃ arises in a most natural way. In fact, what we must understand is exactly which conditions on f ensure that the mapping

c ↦ ∂f^{∗}(c) : Y′ → Y  (1.4)

is norm to norm or at least norm to weak continuous at c, thereby ensuring c_{L} → c is sufficient (and necessary) for the convergence (in some sense) of the sequence of approximate control functions. Unfortunately, the dual norm f^{∗} is only of theoretical value, in as much as it can only be approximated numerically, so we cannot examine its smoothness properties to answer this question. It turns out that the convexity properties of f at the dual solution x̃ guarantee the continuity of the mapping (1.4). Specifically, since Y is reflexive (even if p = ∞), we show that a necessary and sufficient condition for the norm to weak continuity of (1.4) is G-convexity at x̃, which is something closely akin to strict convexity. We go on to show that a necessary and sufficient condition for the norm to norm continuity of (1.4) is F-convexity at x̃. This condition is implied by, but not equivalent to (even in the case of reflexive Banach spaces), local uniform convexity (local uniform rotundity) as defined in [10]. See Section 4 for examples. The next problem one encounters is the inexact knowledge of the eigenvalues themselves, which is the case if the vibrating system under consideration is approximated numerically (e.g. any finite element approximation of the vibrating system). This leads to not only a sequence of approximating right hand sides of (1.1) but also to a sequence of approximating norms {f_{L}}, which incorporate the inexact eigenvalues. This sequence converges uniformly on bounded sets to f. In fact, if we let K be the set of norms on ℓ^{2}(IR^{2}) which are equivalent to f, what we are now asking is what conditions on f ensure that the mapping

(f^{∗}, c) ↦ ∂f^{∗}(c) : K × Y′ → Y  (1.5)

is (uniform cross) norm to norm or at least (uniform cross) norm to weak continuous at (f^{∗}, c). That is, we topologize K with the topology of uniform convergence on bounded sets and give Y′ the norm topology and study the continuity of (1.5) with the norm or weak topology on Y. It turns out that this setting lends itself well to the concepts of G− and F−convexity and that the presence of K in (1.5) changes little in the analysis. Thus, we may exploit the smoothness and convexity of f to once again get that c_{L} → c is the (necessary and) sufficient condition for the norm to weak (G−convexity) or norm to norm (F−convexity) continuity of the mapping (1.5) at (f^{∗}, c), thereby ensuring that c_{L} → c is sufficient (and necessary) for the convergence (in some sense) of the sequence of approximate control functions. This theory makes no demands on the smoothness of the sequence of approximating norms, but asks only that they approximate the limit norm uniformly well on some neighborhood of the point we are interested in. In summary we apply local analysis to examine the duality between convexity in the primal space and differentiability in the dual space, as well as the duality between differentiability in the primal space and convexity in the dual space. In [5] a global theory is developed. We show that the definition of uniform convexity (rotundity) used in the global theory does not lend itself well to a satisfactory local theory. Specifically, for the case p = ∞ we do not have G−convexity at each point, and so one cannot hope to apply global theory (i.e. show f is uniformly convex as in [5]) to examine the continuity of the mapping (1.4) or for that matter (1.5). By relaxing the global setting to a local one, we can apply generic types of results to show that for any c in some dense G_{δ} subset of Y′ we in fact get the much stronger convergence.

Thus, in [13] we are able to apply the theoretical results derived here to extend two earlier works. In [8] approximation methods are employed to approximate (1.3) for p = 2. The geometry of the dual problem is not discussed and no convergence rate is derived. We place these arguments in a geometrical setting which allows the approximation problem to be understood properly. Exploiting the geometry involved we are able to not only give a more general sufficient condition for convergence (c_{L} → c) than that given in [8], but we are also able to show that this condition is also necessary not only for p = 2 but for all p ∈ [2, ∞]. For p ∈ [2, ∞) we show that f is uniformly convex as in [5], allowing stronger convergence results than we get for p = ∞. The case p = ∞ is treated in [9], where calculations are based on truncation and knowledge of the eigenelements. Thus, the finite dimensional subspace used in [9] is span{φ_{j}}_{j=1}^{M}, which is in practice not viable. The powerful geometrical arguments iderived here allow us to weaken significantly the assumptions made (exact knowledge of the eigenelements), while at the same time strengthening the convergence obtained (norm or semi-norm convergence instead of weak^{∗}).

2. Smoothness and the Subdifferential Mapping

We consider a real Banach space Y, on which is defined a convex function f. We will further assume that f is bounded. That is, for each ρ > 0 there is a β_{ρ} > 0 such that if ∥x∥ ≤ ρ, then |f(x)| ≤ β_{ρ}. In other words the function f is bounded on any bounded set V ⊂ Y. This assumption presents no real loss of generality since given any point x ∈ Y, at which a general convex function is continuous, we can construct a bounded convex function which agrees with the general function on some neighborhood of x. This assumption simplifies many of the arguments, so we will retain it throughout.

We will designate the dual of Y by Y′. The subdifferential of f at any x ∈ Y denoted by ∂f(x) ⊂ Y′ and defined by

∂f(x) := {ξ′ ∈ Y′ : ⟨ξ′|y⟩ ≤ f(x + y) − f(x) for all y ∈ Y },  (2.1)

is nonempty, bounded, convex and weak-^{∗} compact [12]. Another result to be gleaned from [12] is that for each x, y ∈ Y we get a nice formula for the directional derivative of f at x in direction y, which we denote by

df(x; y) := lim_{α→0^{+}} \frac{f(x + αy) − f(x)}{α} = max_{ξ′∈∂f(x)} ⟨ξ′|y⟩.

For each x, y ∈ Y we define the function e( · ; x, y) : IR → IR by

e(α; x, y) = \{ \begin{matrix}\frac{f(x + αy) − f(x)}{α} − df(x; y) & if α ≠ 0, \\ 0 & if α = 0, \end{matrix}  (2.2)

which is continuous on IR, except possibly from the left at the origin. For a fixed x ∈ Y ∂f(x) having only one element, which we write ∂f(x) = {∇f(x)}, is equivalent to each function e( · ; x, y) in the family of functions indexed by y ∈ Y being continuous at the origin. But this is the definition of Gateaux differentiability of f at x ∈ Y as in [11].

From the linearity of ⟨ξ′| · ⟩ in (2.1) it is clear that αe(α; x, y) ≥ 0 for all α ∈ IR, and from the convexity of f it follows that e( · ; x, y) is monotonically increasing on IR. We will in fact always choose y on the unit sphere. This choice will not restrict the generality of our arguments, since one can show e(α; x, βy) = βe(αβ; x, y) for all x, y ∈ Y and all β > 0.

In our notation V + W := {x ∈ Y : x = v + w for some v ∈ V and w ∈ W} for any V, W ⊆Y. Further, we let B_{ρ}(x̄) [B_{ρ}] be the open ball of radius ρ > 0 about x̄ [the origin], B̄_{ρ}(x̄) [B̄_{ρ}] be the closed ball of radius ρ ≥0 about x̄ [the origin] and S_{ρ}(x̄) [S_{ρ}] be the sphere of radius ρ > 0 about x̄ [the origin]. We will define the multiplication of a set by a scalar in the classical way αV := {x ∈ Y : x = αy for some y ∈ V}.

We let C_{0}(IR) be the set of all continuous, strictly monotonically increasing functions on IR which vanish at the origin. Recall that if ϕ is a real valued, convex function on some open interval I ⊆ IR, then we can define the right- and left-hand derivatives D^{±}ϕ of ϕ on I, which are monotonically increasing functions, and satisfy D^{−}ϕ(τ) := min_{∂ϕ(τ)} ≤ max_{∂ϕ(τ)} =: D^{+}ϕ(τ) for all τ ∈ I.

2.1. Differentiability Concepts

In order to give a concise yet general concept of the differentiability of a bounded, convex function f, we consider again e given by (2.2) and two nonempty sets V ⊆ Y and W ⊆ S_{1}. We will take a bornological point of view here and in the next section.

Definition 2.1. We say e admits a uniform estimate [to the right] on V with respect to W if there exists E_{W} ∈ C_{0}(IR) such that |e(α; x, y)| ≤ |E_{W}(α)| for all y ∈ W, all x ∈ V and all α ∈ IR [α ≥ 0]. We also say E_{W} estimates e uniformly well [to the right] on V with respect to W. If V = {x_{0}} we say e admits an estimate [to the right] at x_{0} with respect to W.

There are three classes of subsets of S_{1} which interest us. They are G, the class of all finite subsets of S_{1}, H, the class of all compact subsets of S_{1}, and F, the class of all bounded subsets of S_{1}.

Definition 2.2. If e admits a uniform estimate [to the right] on V with respect to each

W ∈ \{ \begin{matrix}G, then we say that f is uniformly G − [G^{+}−]differentiable on V. \\ H, then we say that f is uniformly H − [H^{+}−]differentiable on V. \\ F, then we say that f is uniformly F − [F^{+}−]differentiable on V. \end{matrix}

If in this definition V = {x}, we say that f is differentiable at x in the sense under consideration. In as much as we are considering functons which are bounded and convex, it is easy to see that in our terminology (uniform) G−differentiability is exactly (uniform) Gateaux differentiability, and (uniform) F−differentiability is exactly (uniform) Fréchet differentiability. We also remind the reader that due to the Lipschtz continuity of the functions under consideration H − [H^{+}−]differentiability is subsumed in the concept of G − [G^{+}−]differentiability. We will need this result later. See [1].

Since S_{1} ∈ F, we have that if f is uniformly F−differentiable on V ⊆ Y, there exists some E ∈ C_{0}(IR), such that for all x ∈ V, all α ∈ IR and all y ∈ S_{1}, we get

f(x + αy) ≤ f(x) + α⟨∇f(x)|y⟩ + αE(α).

Lemma 2.3. Let f be a bounded, convex function over the Banach space Y. Then the set ∪_{x∈V}∂f(x) is norm bounded if V is.

We remind the reader of the following variation of the Mean Value Theorem, the proof of which can be found in [1].

Lemma 2.4. If f is bounded and convex on the Banach space Y, then for x, y ∈ Y there is some τ_{0} ∈ [0, 1] and some ξ′ ∈ ∂f(x + τ_{0}y) such that f(x + y) − f(x) = ⟨ξ′|y⟩.

2.2. Differentiability and Continuity of the Subdifferential Mapping

The set of all bounded convex functions forms a convex cone K in the vector space IR^{Y}. By fixing V ⊂ Y and ρ > 0, we can endow K × Y with the semi-metric σ generated by the semi-norm on K given by ∥f∥_{V,ρ} := sup_{x∈V+B_{ρ}} |f(x)| and the norm on Y. Thus,

σ ((f, x), (f_{0}, x_{0})) := max{∥f − f_{0}∥_{V,ρ} , ∥x − x_{0}∥}.  (2.4)

For each δ > 0 we say (f, x) ∈ B_{δ}((f_{0}, x_{0})) if σ((f, x), (f_{0}, x_{0})) < δ. Using this topology on K × Y, we want to explore the continuity of the mapping

(f, x) ↦ ∂f(x) : K × Y → C,  (2.5)

where C ⊂ 2^{Y′} is the set of all non-empty, convex, weak-^{∗} compact subsets of Y′. In a subsequent paper we will consider various hyper-topologies on C, each of which has very interesting implications for set convergence. This would take us too far afield, so in this paper we will define what continuity means directly in terms of elements of the subdifferentials. In the subsequent paper these definitions will then become conditions equivalent to the type of continuity under consideration there. Now we define the two types of [upper semi-]continuity that we are interested in. The reader should keep the example of the norm in mind. It is well known that for the norm the G− [uniform F−]differentiability of the norm at a point of the unit sphere [on V ⊆ S_{1}] is equivalent to the norm to weak^{∗} [norm to norm uniform] continuity of any selection of the subdifferential mapping [5]. That is, if {x_{L}} ⊂ S_{1} and x_{L} → x_{0}, then we get ξ′_{L}⇀^{⋆}ξ′_{0} [ξ′_{L} → ξ′_{0} uniformly for x_{0} ∈ V], where ξ′_{L} is any selection from ∂f(x_{L}) and ξ′_{0} is any selection (the gradient) from ∂f(x_{0}). We generalize these concepts in the following definitions.

Definition 2.5. Mapping (2.5) is uniformly G−[F−]upper semi-continuous on {f_{0}}×V if for ε > 0 and W ∈ G [F] there exists a δ > 0 (independent of x_{0} ∈ V) such that if x_{0} ∈ V, if (f, x) ∈ B_{δ}((f_{0}, x_{0})) and if ξ′ ∈ ∂f(x), then sup_{y∈W}( ⟨ξ′|y⟩ − df_{0}(x_{0}; y) ) < ε.

Definition 2.6. Mapping (2.5) is uniformly G −[F−]continuous on {f_{0}}×V if for each ε > 0 and W ∈ G [F] there exists a δ > 0 (independent of x_{0} ∈ V) such that if x_{0} ∈ V, if (f, x) ∈ B_{δ}((f_{0}, x_{0})), if ξ′ ∈ ∂f(x) and if ξ′_{0} ∈ ∂f_{0}(x_{0}), then sup_{y∈W} |⟨ξ′ − ξ′_{0}|y⟩| < ε.

If V = {x_{0}}, then we say the mapping is G− or F−[upper semi-]continuous at x_{0}.

We remind the reader that S_{1} ∈ F, so that one works with exactly this set in that setting. We also note that G− and F− continuity of (2.5) at x_{0} imply that ∂f_{0}(x_{0}) is a singleton. This should come as no suprise, as we can characterize the continuity of the mapping (f, x) ↦ ∂f(x) : K × Y → C in terms of the smoothness of f. This characterization is our next goal. One could in fact obtain a uniform version of the next result by using uniform G − [G^{+}−]differentiability and uniform G−[upper semi-]continuity. Such a result is, however, of little practical interest for us, as this entails being able to estimate e uniformly in x ∈ V for each fixed W ∈ G. Instead, we go on to give a very useful uniform version of F−[upper semi-]continuity. The next proof is simpler and the results are somewhat stronger than the more traditional formulation of upper semi-continuity in [11].

Theorem 2.7. The mapping (f, x) ↦ ∂f(x) : K×Y → C is G−[upper semi-]continuous at (f_{0}, x_{0}) ∈ K × Y if and only if f_{0} is G − [G^{+}−]differentiable at x_{0}.

Proof. Assume first the G − [G^{+}−]differentiability of f_{0} at x_{0}. We fix ε > 0 and W ∈ G. Then there exists E_{W} ∈ C_{0}(IR) such that |e(α; x_{0}, y)| ≤ |E_{W}(α)| for all y ∈ W and all α ∈ IR [α > 0]. Fix ρ > 0, and let γ ≥ 1 be a Lipschtz constant for f on B_{ρ}(x_{0}). Choose α ∈ (0, ρ) small enough that E(α) < ε/3. Define δ := min{ρ − α, αε/6γ}. Assume that (f, x) ∈ B_{δ}((f_{0}, x_{0})) and that ξ′ ∈ ∂f(x). Fix y ∈ W. Now let ξ′_{y} = ∇f_{0}(x_{0}) [ ξ′_{y} ∈ ∂f_{0}(x_{0}) satisfy ⟨ξ′_{y}|y⟩ = df(x; y) for our fixed y ∈ W ]. Upon rearranging (2.2) with f_{0} and x_{0} and using (2.1) we get the inequality

⟨ξ′ − ξ′_{y}|y⟩ ≤ \frac{f(x + αy) − f(x)}{α} − \frac{f_{0}(x_{0} + αy) − f_{0}(x_{0})}{α} + e(α; x, y)
≤ \frac{f(x + αy) − f_{0}(x + αy)}{α} + \frac{f_{0}(x) − f(x)}{α}+
\frac{f_{0}(x + αy) − f_{0}(x_{0} + αy)}{α} + \frac{f_{0}(x_{0}) − f_{0}(x)}{α} + E_{W}(α)  (2.6)
≤ 2δ/α + 2γ∥x − x_{0}∥/α + E_{W}(α)
< ε.

Since this holds for each y ∈ W, the definition of G−upper semi-continuity of mapping (2.5) at x_{0} is satisfied. Moreover, if f_{0} is G-differentiable at x_{0}, then ∂f_{0}(x_{0}) = {∇f_{0}(x_{0})}, and we can use similar arguments, with α ∈ (−ρ, 0) to obtain the estimate −ε < ⟨ξ′−ξ′_{y}|y⟩ for each y ∈ W, so that (2.5) is G−continuous at x_{0}.

Now assume that the mapping (f, x) ↦ ∂f(x) : K × Y → C is G−continuous at (f_{0}, x_{0}). Fix W ∈ G. Choose y ∈ W, and let ξ′_{y} ∈ ∂f_{0}(x_{0}) satisfy ⟨ξ′_{y}|y⟩ = df(x; y). Then upon rearranging (2.2) with ⟨ξ′_{y}|y⟩ replacing df(x_{0}; y) and applying Lemma 2.4, we get

e(α; x_{0}, y) = ⟨ξ′_{α} − ξ′_{y}|y⟩  (2.7)

for some τ ∈ [0, 1] and some ξ′_{α} ∈ ∂f_{0}(x_{0} + ταy). Clearly, the G−continuity of (2.5) implies that e(α; x_{0}, y) = ⟨ξ′_{α} − ξ′_{y}|y⟩ → 0 as α → 0 for each y ∈ W. Let E : IR → IR be defined by

E_{W}(α) := sgn(α) · max_{y∈W} |e(α; x_{0}, y)| + α.  (2.8)

Then we have that E_{W} vanishes continuously at the origin so that E_{W} ∈ C_{0}(IR), so that we get E_{W} estimates e at x_{0} with respect to W. Since this construction can be carried out for each W ∈ G, we have by definition that f_{0} is G−differentiable at x_{0}. Construction (2.8) gives us the G^{+}-differentiability for free, since W ∈ G and since for each y ∈ W we get e( · ; x_{0}, y) is monotonically increasing and continuous from the right at the origin. In this case we define E_{W} : IR → IR by (2.8) for α ≥ 0 and by E_{W}(α) = α for α < 0.

Corollary 2.8. Mapping (2.5) is G −(H−)upper semi-continuous at any (f, x) ∈ K×Y.

We can exploit uniformity in y ∈ S_{1} to extend Theorem 2.7 to get

Theorem 2.9. Let f_{0} be a bounded, convex function on Y with V ⊆ Y and ρ > 0. Consider the following conditions.

(i) f_{0} is uniformly F − [F^{+}−]differentiable on V and ∪_{x∈V+B_{ρ}}∂f_{0}(x) is bounded. (ii) Mapping (2.5) is uniformly F−[upper semi-] continuous on {f_{0}} × V. (iii) f_{0} is uniformly F − [F^{+}−]differentiable on V. Then we have (i)⇒(ii)⇒(iii); however, (i) and (ii) are not equivalent.

Proof. The proof that (i) ⇒ (ii) follows by modifying the arguments leading up to the inequalities in (2.6). Using the uniform estimate in (2.3) that we get on e, we can replace W ∈ G everywhere by S_{1}. In (2.6) we can estimate both

\frac{f_{0}(x_{0}) − f_{0}(x)}{α}  and  \frac{f_{0}(x + αy) − f_{0}(x_{0} + αy)}{α}

by using Lemma 2.4 and the boundedness of ∪_{x∈V+B_{ρ}}∂f_{0}(x). This gives us the inequalies in (2.6) for each y ∈ S_{1}, so that by taking the supremum over y ∈ S_{1} we get the definitions (modulo the equality that the supremum might give) required in (ii). The proof that (ii)⇒(iii) is also an immediate extension of Theorem 2.7.

To see that (ii)⇏(i) in general, it suffices to consider f : IR → IR where f(x) = x^{2}/2. Then f(x + α) − f(x) = xα + α^{2}/2 so that f is uniformly F−differentiable on IR. The mapping (f, x) ↦ ∂f(x) = {x} is uniformly F−continuous on IR, but ∪_{x∈IR}∂f(x) is not bounded there. Boundedness of V is sufficient to ensure that (i) and (ii) are equivalent.

3. Convexity and the “Inverted” Subdifferential Mapping

In the previous section we considered the mapping (f, x) ↦ ∂f(x) : K × Y → C and saw that various types of continuity of this mapping at the point (f, x) in fact characterize the smoothness of f at x ∈ Y. Now for a fixed f ∈ K and Z ⊆ Y the mapping ∂f : Z → C defines a binary relation

R_{f,Z} := graph(∂f) ∩ (Z × Y′)
= {(x, ξ′) ∈ Z × Y′ : x ∈ Z and ξ′ ∈ Y′ with ξ′ ∈ ∂f(x)}
= ∪_{x∈Z} {x} × ∂f(x)

If we define Z′ := ∪_{x∈Z}∂f(x) ⊆ Y′ and let f^{∗} be the conjugate of f, we see that the inverse relation

R_{f,Z}^{−1} := graph(∂f^{∗}) ∩ (Y′ × Z)
= {(ξ′, x) ∈ Z′ × Z : ξ′ ∈ Z′ and x ∈ Z with ξ′ ∈ ∂f(x)}
= ∪_{ξ′∈Z′} {ξ′} × {x ∈ Z : ξ′ ∈ ∂f(x)}

defines implicitly a multivalued function ∂f_{Z}^{∗} : Z′ → 2^{Z}. Our aim in this section is to develop a theory which describes the continuity of the mapping

(f, ξ′) ↦ ∂f_{Z}^{∗}(ξ′) : (K × Z′) → 2^{Z},  (3.1)

and which parallels the theory developed in the last section. That is, given the appropriate concepts of continuity, we want to relate certain properties of f to the continuity of (3.1) at (f, ξ′) or more generally on {f} × V′ for some V′ ⊆ Z′. Once again our vehicle will be equation (2.2). In order to develop a more broadly applicable theory we consider Z ⊆ Y, defined in terms of f, to be one of three examples. These are Y, the domain of f, as well as the level and tangent sets of f at some x_{0}, where

Definition 3.1. The level set of f is L_{α} := {x ∈ Y : f(x) = α}.

Definition 3.2. The tangent set of f at x_{0} is T_{f}(x_{0}) := {x ∈ Y : df(x_{0}; x − x_{0}) = 0}. We wish to study the geometry of the set R_{f,Z} at the point x_{0} ∈ Z, so we assume that x_{0} and Z have been chosen so that

ρ_{x_{0}} := sup{ρ ≥ 0 : Z ∩ S_{α}(x_{0}) ≠ ∅ for all α ∈ [0, ρ]} ∈ (0, ∞].  (3.2)

Definition 3.3. In (3.2) we call ρ_{x_{0}} the diameter of Z at x_{0}, and we write dia(Z_{x_{0}}) = ρ_{x_{0}}.

If Z = Y then ρ_{x_{0}} = ∞ for each x_{0} ∈ Y. Further, if Z = T_{f}(x_{0}) and 0 ∉ ∂f(x_{0}), then (assuming dim(Y) ≥ 2) we get ρ_{x_{0}} = ∞. Finally, if Z = L_{f(x_{0})} and ρ_{x_{0}} = 0, then x_{0} is the unique minimum of f. Thus, if we take Z to be either of these last two cases, we make the assumption that 0 ∉ ∂f(x_{0}). We define the family of subsets of the unit sphere

\begin{matrix}Z_{α}(x_{0}) := & \{y ∈ S_{1} : x_{0} + αy ∈ Z\} \\ = & (1/α) [ (Z ∩ S_{α}(x_{0})) + \{−x_{0}\} ]. \end{matrix}

For notational reasons let Z_{α}(x_{0}) = ∅ for α ≥ ρ_{x_{0}}. Thus, if Z = Y, then Z_{α}(x_{0}) = S_{1} for all α > 0. If Z = T_{f}(x_{0}), then Z_{α}(x_{0}) = T_{f}(x_{0}) ∩ S_{1}(x_{0}) + {−x_{0}} for all α > 0.

3.1. Convexity Concepts and the Bregman Distance Function

In order to give a concise yet general concept of the convexity of a bounded, convex function f, we consider again e given by (2.2) and two nonempty sets V ⊆ Z and W ⊆ S_{1}. If we set inf ∅ = ∞, then for each W ⊆ S_{1} and x ∈ V we can define the increasing function (which is similar to the Bregman distance function) μ_{x,W} : [0, ∞) → [0, ∞] by

μ_{x,W}(α) := \{ \begin{matrix}0 & if α = 0 \\ inf_{β≥α}\{e(β; x, y) : y ∈ W ∩ Z_{β}(x)\} & if α > 0 \end{matrix}

Definition 3.4. We say that e admits a Z−defined (lower) estimate on V with respect to W if there is some F_{W} ∈ C_{0}(IR) such that μ_{W}(α) := inf_{x∈V} μ_{x,W}(α) ≥ F_{W}(α) for each α > 0. The Z−defined modulus of convexity of f on V is μ_{S_{1}}, which we write μ. If V = {x}, we say e admits a Z−defined (lower) estimate at x with respect to W.

Example 3.5. If f is the norm on a Hilbert space, then the modulus of convexity at any point of the unit sphere defined in terms of the unit sphere is μ(α) = α/2.

Definition 3.6. If e admits a Z−defined (lower) uniform estimate on V with respect to each

W ∈ \{ \begin{matrix}G, then f is uniformly G−convex on V with respect to Z. \\ H, then f is uniformly H−convex on V with respect to Z. \\ F, then f is uniformly F−convex on V with respect to Z. \end{matrix}

If in this definition V = {x}, we say that f is convex at x in the sense under consideration. Note, too, that if Z = T_{f}(x_{0}), then we always take V = {x}.

Lemma 3.7. The G−convexity of f at x_{0} with respect to Z (for each Z) is sufficient to ensure that L_{f(x_{0})} ∩ T_{f}(x_{0}) = {x_{0}}. It is also necessary if Z ≠ Y.

Proof. Assume f is G−convex at x_{0} with respect to Z. Fix x ∈ L_{f(x_{0})} ∩ T_{f}(x_{0}) and set ᾱ := ∥x − x_{0}∥. Choose y ∈ S_{1} such that ᾱy = x − x_{0}. Then x_{0} + αy ∈ T_{f}(x_{0}) for all α ∈ [0, ᾱ]. If ᾱ > 0, then df(x_{0}; y) = 0, and e(α; x_{0}, y) = 0 for all α ∈ [0, ᾱ], contradicting the G−convexity of f at x_{0}. Thus ᾱ = 0 and L_{f(x_{0})} ∩ T_{f}(x_{0}) = {x_{0}} as required. Note that we make no use of the assumption 0 ∉ ∂f(x_{0}) up to this point. Let L_{f(x_{0})} ∩ T_{f}(x_{0}) = {x_{0}}. If Z = T_{f}(x_{0}), then for each α > 0 and y ∈ Z_{α}(x_{0}) we have f(x_{0} +αy) > f(x_{0}), so that αe(α; x_{0}, y) = f(x_{0} +αy)−f(x_{0}) > 0. Thus, since W ∈ G, we can define F_{W}(α) := inf_{y∈W} e(α; x_{0}, y) ∈ C_{0}(IR) for α ≥ 0 and F_{W}(α) := −α for α < 0. Then F_{W} estimates μ_{x_{0},W} (below) at x_{0} with respect to T_{f}(x_{0}). If Z = L_{f(x_{0})}, then for each α > 0 we get [Z_{α}(x_{0})+{x_{0}}]∩T_{f}(x_{0}) = ∅, so that y ∈ Z_{α}(x_{0}) implies −df(x_{0}; y) > 0. Thus, inf_{β≥α}{e(β; x, y) : y ∈ W ∩ Z_{β}(x_{0})} = inf_{β≥α}{−df(x_{0}; y) : y ∈ W ∩ Z_{β}(x_{0})} > 0. Since W ∈ G, this is positive for each α > 0, so that we can find some F_{W} ∈ C_{0}(IR) which estimates μ_{x_{0},W} (below) at x_{0} with respect to T_{f}(x_{0})

Thus, we see that the G− convexity of f at x_{0} with respect to L_{f(x_{0})} is equivalent to the G− convexity of f at x_{0} with respect to T_{f}(x_{0}).

Note that in both cases it is essential that Z ≠ {x_{0}} so that f not be trivially G−convex at x_{0} with respect to Z. Thus, 0 ∉ ∂f(x_{0}) could be weakened to Z ≠ {x_{0}}. Note, too, that if 0 ∈ ∂f(x_{0}), then L_{f(x_{0})} ⊆ T_{f}(x_{0}), so L_{f(x_{0})} ≠ {x_{0}} if and only if L_{f(x_{0})} ∩ T_{f}(x_{0}) ≠ {x_{0}}. Of course, the conditions 0 ∈ ∂f(x_{0}) and L_{f(x_{0})} = {x_{0}} together are the same as requiring that x_{0} is the unique minimum of f. Thus, if 0 ∈ ∂f(x_{0}), then the G−convexity of f at x_{0} with respect to L_{f(x_{0})} or T_{f}(x_{0}) is equivalent to f having a unique minimum at x_{0}. As is the case if f is a norm, if 0 ∈ ∂f(x_{0}), then f can be (trivially) G−convex with respect to L_{f(x_{0})} but not with respect to Y.

If Z = Y we do not expect necessity in Lemma 3.7. To see this, consider the Euclidian norm on IR^{2} with x_{0} = (1, 0). Then L_{f(x_{0})} is the unit sphere and T_{f}(x_{0}) = {(1, τ) : τ ∈ IR}. Thus, we get that L_{f(x_{0})} ∩ T_{f}(x_{0}) = {x_{0}}. That f in this example is not G−convex at x_{0} with respect to IR^{2} follows from

Lemma 3.8. If Z = Y, then G−convexity at x_{0} is exactly strict convexity at x_{0} as expressed by the strict inequality

f(λx_{0} + (1 − λ)(x_{0} + y)) < λf(x_{0}) + (1 − λ)f(x_{0} + y)  (3.3)

for all λ ∈ (0, 1) and all y ∈ Y \{0}.

Proof. By setting α = 1 − λ and β = ∥y∥ we see that (3.3) is equivalent to the strict inequality e(βα; x_{0}, y) < e(β; x_{0}, y) for all α ∈ (0, 1), for all β > 0 and for all y ∈ S_{1}. Thus, strict convexity at x_{0} implies e(α; x_{0}, y) > 0 for all α > 0, so that given W ∈ G we can construct a (lower) estimate F_{W} ∈ C_{0}(IR) for μ_{x_{0},W} just as in Lemma 3.7.

If on the other hand f is not strictly convex at x_{0}, then we can find some λ ∈ (0, 1) and y ≠ 0 such that equality holds in (3.3). Convexity then implies that for this y we get equality for all λ ∈ [0, 1]. This means that e(α; x_{0}, y/∥y∥) = 0 for all α ∈ [0, ∥y∥], violating the G−convexity of f at x_{0} with respect to Y.

In analog to the previous section, it is interesting to note the equivalence of G− and H−convexity with respect to Z. We also note that the Z−defined modulus of convexity of f on V is monotonically increasing, since μ_{x,S_{1}} is monotone for x ∈ V and μ := inf_{x∈V} μ_{x,S_{1}}.

Lemma 3.9. The Z−defined modulus of convexity of f on V is continuous at the origin.

Proof. Note that in as much as μ := inf_{x∈V} μ_{x,S_{1}}, it suffices to show this at a point. The results follow from monotonicity if f is not F−convex at x_{0}, so we assume it is. First we let Z = Y or Z = T_{f}(x_{0}). Then Z_{α}(x_{0}) is independent of α > 0, so we can write 0 ≤ μ_{x_{0},S_{1}}(α) ≤ μ_{x_{0},{y}}(α) := e(α; x_{0}, y) for any y ∈ Z_{α}(x_{0}), so that the results follow by the continuity of e( · ; x_{0}, y) at the origin.

Now assume Z = L_{f(x_{0})}. Choose ỹ ∈ S_{1} such that x_{0} + ỹ ∈ T_{f}(x_{0}). Choose ŷ ∈ S_{1} such that x_{0} + α̂ŷ ∈ Zα̂(x_{0}) for some α̂ ∈ (0, ρ_{x_{0}}). Then for each α ∈ [0, α̂] we can find some y_{α} ∈ W := S_{1} ∩ {x : x = ν̂ŷ + ν̃ỹ where ν̂, ν̃ ≥ 0} such that x_{0} + αy_{α} ∈ L_{f(x_{0})}. Since W ∈ H, we can find some E_{W} ∈ C_{0}(IR) which estimates e (above) with respect to W. From this we get the estimate μ_{x_{0},S_{1}}(α) ≤ df(x_{0}; y_{α}) =: e(α; x_{0}, y_{α}) ≤ E_{W}(α) which means that μ_{x_{0},S_{1}} is continuous at the origin. Note that here 0 ∉ ∂f(x_{0}) is assumed.

As a consequence of this lemma and the monotonicity of μ, if f is uniformly F−convex on V, then for each α ≥ 0 we can define μ̂(α) := αμ(α), which is strictly increasing in α ∈ IR^{+} and continuous at the origin. Thus, we can define μ̂^{−1} : IR^{+} → IR^{+} by

μ̂^{−1}(β) := α for all β ∈ [ lim_{γ→α^{−}} μ̂(γ), lim_{γ→α^{+}} μ̂(γ)],  (3.4)

if for notational convenience, we agree that lim_{γ→0^{−}} μ(γ) = 0. Clearly μ̂^{−1} is positive on (0, ∞), is monotonically increasing on IR^{+} and is continuous at the origin with μ̂^{−1}(0) = 0.

3.2. Convexity and Continuity of the Inverse Mapping

We observe that for x_{0}, x_{1} ∈ Z and ξ′ ∈ Z′ we have the equivalence of x_{0}, x_{1} ∈ ∂f_{Z}^{∗}(ξ′) and ξ′ ∈ ∂f(x_{0}) ∩ ∂f(x_{1}). Thus, the cardinality of ∂f_{Z}^{∗}(ξ′) being one for any ξ′ ∈ ∂f(x_{0}) is equivalent to the condition ∂f(x_{0}) ∩ ∂f(x_{1}) = ∅ for all x_{1} ∈ Z\{x_{0}}. This leads us to the dual equivalent to the statement that the function f is G−differentiable at a point if and only if the subdifferential mapping is single valued there.

Theorem 3.10. The cardinality of the inverse mapping (3.1) is one on {f} × ∂f(x_{0}) if and only if f is G-convex at x_{0} with respect to Z.

Proof. In the light of Lemma 3.8 this is classical for Z = Y [3]. Thus we consider only the case Z = L_{f(x_{0})} (and hence Z = T_{f}(x_{0})). We assume 0 ∉ ∂f(x_{0}) throughout.

First assume f is G-convex at x_{0} with respect to Z = L_{f(x_{0})}. Let ξ′ ∈ ∂f(x_{0}) ∩ ∂f(x) for some x ∈ Z. Define the function ϕ by ϕ(τ) := f(x_{0} + τ(x − x_{0})) for each τ ∈ IR. Since ϕ(0) = ϕ(1), we get D^{+}ϕ(0) ≤ 0 ≤ D^{−}ϕ(1). Now, the fact that ξ′ ∈ ∂f(x_{0})∩∂f(x_{1}) gives the inequalities D^{−}ϕ(1) ≤ ⟨ξ′|x − x_{0}⟩ ≤ D^{+}ϕ(0). Thus, monotonicity of D^{+}ϕ means that D^{+}ϕ ≡ 0 on [0, 1), so that x ∈ T_{f}(x_{0}), and hence x = x_{0} by Lemma 3.7.

Now assume the cardinality of the inverse mapping (3.1) is one on {f} × ∂f(x_{0}). Let x ∈ L_{f(x_{0})} ∩ T_{f}(x_{0}). Choose ξ′_{0} ∈ ∂f(x_{0}) such that ⟨ξ′_{0}|x − x_{0}⟩ = df(x_{0}; x − x_{0}) = 0. For all y ∈ Y we can write

\begin{matrix}⟨ξ′_{0}|y⟩ = & ⟨ξ′_{0}|x + y − x_{0}⟩ \\ ≤ & f(x + y − x_{0} + x_{0}) − f(x_{0}) \\ = & f(x + y) − f(x). \end{matrix}

Thus, ξ′_{0} ∈ ∂f(x) by definition. Since x ∈ Z, single valuedness of (3.1) implies x = x_{0}. But this is true for all x ∈ L_{f(x_{0})} ∩ T_{f}(x_{0}), so that by Lemma 3.7 we get that f is G−convex at x_{0} with respect to Z.

Unfortunately, the dual equivalent of Theorem 2.7 is not so readily availible. The temptation is to replace weak-^{∗} concepts in Y′ by their weak analogs in Y, and thereby be able to characterize the G−continuity of the inverse mapping (3.1) in terms of the G−convexity of f at a point. It turns out that G−convexity is too weak for this to be sucessful. We now state the definition of G −[F−]continuity of mapping (3.1) and show that this is sufficient for the G−convexity at a point. We then go on to show that in the proper setting this is necessary, too.

Fix f_{0} ∈ K. Let x_{0} ∈ V_{0} ⊆ Z_{0} and define V′_{0} := ∪_{x∈V_{0}}∂f_{0}(x). Similarly for some f ∈ K let V ⊆ Z and define V′ := ∪_{x∈V}∂f(x). We also define the families G′ and F′ of subsets of the unit sphere S′_{1} in Y′ in the obvious way. Note that in (2.4) we make use of a set V, which is not necessarily the same V as here. We will always make clear what V in (2.4) is.

Definition 3.11. Mapping (3.1) is uniformly G − [F−]continuous on {f_{0}} × V′_{0} if for each ε > 0 and W′ ∈ G′ [F′] there exists a δ > 0 (independent of ξ′_{0} ∈ V′_{0}) such that if ξ′_{0} ∈ V′_{0}, if (f, ξ′) ∈ B_{δ}((f_{0}, ξ′_{0})), if x ∈ ∂f_{Z}^{∗}(ξ′), if x_{0} ∈ ∂f_{0}^{∗}_{Z_{0}}(ξ′_{0}) and if |f(x)−f_{0}(x_{0})| < δ, then we get that sup_{η′∈W′} |⟨η′|x − x_{0}⟩| < ε.

Note that the Euclidian norm on IR^{2} is G−convex at any point x_{0} of the unit sphere with respect to the unit sphere, but this will not guarantee the continuity of (3.1) without the condition |f(x) − f_{0}(x_{0})| < δ. As we mentioned, the G−continuity of (3.1) ensures the G−convexity of f. We now prove this fact.

Theorem 3.12. If the inverse mapping (3.1) is (uniformly) G−continuous on {f} × ∂f(x_{0}) for some x_{0} ∈ Z, then f is G−convex at x_{0} with respect to Z.

Proof. First we assume Z = L_{f(x_{0})} or Z = T_{f}(x_{0}). Fix x ∈ L_{f(x_{0})} ∩ T_{f}(x_{0}). Choose ξ′ ∈ S′_{1} such that ⟨ξ′|x − x_{0}⟩ = ∥x − x_{0}∥ and ξ′_{0} ∈ ∂f(x_{0}) with ⟨ξ′_{0}|x − x_{0}⟩ = 0. We can show (as in Lemma 3.10) that ξ′_{0} ∈ ∂f(x). Thus, the G−continuity of (3.1) implies, upon choosing W′ = {ξ′}, that ∥x − x_{0}∥ < ε for all ε > 0 so that L_{f(x_{0})} ∩ T_{f}(x_{0}) = {x_{0}}. At this point Lemma 3.7 gives the results.

Now let Z = Y. Fix ȳ ∈ S_{1} and assume e(α; x_{0}, ȳ) = 0 for all α ∈ [0, ᾱ], where ᾱ ≥ 0. Let ξ′_{0} ∈ ∂f(x_{0}) satisfy ⟨ξ′_{0}|ȳ⟩ = df(x_{0}; ȳ). Using the definition of ξ′_{0} ∈ ∂f(x_{0}) and of e we get the inequality

\begin{matrix}⟨ξ′_{0}|y⟩ ≤ & f(x_{0} + y) − f(x_{0}) \\ = & f(x_{0} + αȳ + y − αȳ) − f(x_{0} + αȳ) + f(x_{0} + αȳ) − f(x_{0}) \\ = & f(x_{0} + αȳ + y − αȳ) − f(x_{0} + αȳ) + αdf(x_{0}; ȳ), \end{matrix}

which is valid for all y ∈ Y and all α ∈ [0, ᾱ]. Thus, if we set ỹ = y − αȳ, we get by definition that ξ′_{0} ∈ ∂f(x_{0} + αȳ) for each α ∈ [0, ᾱ]. By G−continuity we get ᾱ = 0. This gives us the G−convexity of f at x_{0} with respect to Y.

The next example indicates that G−convexity alone is too weak to assure the G−continuity of mapping (3.1).

Example 3.13. Let f : ℓ^{2} → IR be given by f(x) := ⟨e_{1}, x⟩ + ∑_{j=1}^{∞} \frac{1}{j}(x_{j})^{2} for

x = (x_{j})_{j∈IN} ∈ ℓ^{2}. It is easy to show that f is a bounded, convex function, which is G−convex at the origin with respect to L_{0}. However, if we let x_{L} = (\sqrt{1/4 − L^{−1/2}} − 1/2)e_{1} + L^{1/4}e_{L} ∈ L_{0}, we see that x_{L} ⇀̸ 0. This together with the fact that

∇f(x_{L}) = (\sqrt{1/4 − L^{−1/2}} + 1/2)e_{1} + L^{−3/4}e_{L} → e_{1} = ∇f(0)

gives us a contradiction to the sufficency of G−convexity for the weak continuity of mapping (3.1) in the dual equivalent of Theorem 2.7 even in the Hilbert space ℓ^{2}. The problem here being that the level sets of f are not bounded. In fact, this alone (even on a reflexive space) is not enough as a variation on the last example shows.

Example 3.14. Let f_{L} : ℓ^{2} → IR be given by f_{L}(x) := max\{∑_{j=1}^{L} \frac{1}{j}(x_{j})^{2}, ∥x∥ − L\}

and let f_{0}(x) := max\{∑_{j=1}^{∞} \frac{1}{j}(x_{j})^{2}, ∥x∥ − 1\}. We see that f_{0} is G−convex at the origin

and that f_{L} → f_{0} uniformly on the closed unit ball. If we set x_{0} = 0 and x_{L} := L^{1/4}e_{L} for each L ∈ IN, then f_{L}(x_{L}) → 0 = f_{0}(x_{0}) and ∇f_{L}(x_{L}) = 2L^{−3/4} → 0 = ∇f_{0}(x_{0}). It is however clear that x_{L} ⇀̸ x_{0}.

What we actually need is some condition that ensures that {x_{L}} is bounded (Y reflexive) and all of its weak cluster points are in the set L_{f_{0}(x_{0})} ∩ T_{f_{0}}(x_{0}). Then Lemma 3.7 gives the results. The following notation will be useful.

Definition 3.15. The α-union of f is U_{α} := {x ∈ Y : f(x) ≤ α}.

One way to accomplish the goals outlined above is

Theorem 3.16. Let f_{0} be a bounded convex function with bounded level sets defined on the reflexive Banach space Y. Given a sequence of bounded convex functions {f_{L}} and a null sequence {δ_{L}} such that |f_{L}(x)−f_{0}(x)| < δ_{L} for all x ∈ U_{α}, where α > f_{0}(x_{0}), assume that there is {x_{L}} with ξ′_{L} ∈ ∂f_{L}(x_{L}) such that ξ′_{L} → ξ′_{0} ∈ ∂f_{0}(x_{0}). If f_{L}(x_{L}) → f_{0}(x_{0}), say |f_{L}(x_{L}) − f_{0}(x_{0})| ≤ δ_{L}, and if f_{0} is G−convex at x_{0} with respect to Z, then x_{L} ⇀ x_{0}.

Proof. Let α_{L} := ∥x_{L} − x_{0}∥ and choose y_{L} ∈ S_{1} such that α_{L}y_{L} = x_{L} − x_{0}. By the boundedness of the level sets of f_{0} we can let β̄ := sup_{x∈L_{α}} ∥x − x_{0}∥. If we choose β_{L} > 0 such that x_{0} + β_{L}y_{L} ∈ L_{α} then we get the inequality

\begin{matrix}\frac{α − f_{0}(x_{0}) − 2δ_{L}}{β̄} ≤ & \frac{f_{L}(x_{0} + β_{L}y_{L}) − f_{L}(x_{0})}{β_{L}} \\ ≤ & \frac{f_{L}(x_{L}) − f_{L}(x_{0})}{α_{L}} → 0 \end{matrix}

for all L ∈ IN such that α_{L} ≥ β_{L}. Thus, there are only finitely many L such that x_{L} ∉ U_{α}.

Now let x̄ ∈ U_{α} be any weak cluster point of {x_{L}}. Then since |f_{L}(x_{L}) − f_{0}(x_{L})| ≤ δ_{L} for all but finitely many L, we can write that f_{0}(x̄) ≤ lim inf f_{0}(x_{L}) = f_{0}(x_{0}). Thus, x̄ ∈ U_{f_{0}(x_{0})}. This means that ⟨ξ′|x̄ − x_{0}⟩ ≤ 0 for all ξ′ ∈ ∂f_{0}(x_{0}). On the other hand, from (2.1) we get ⟨ξ′_{L}|x_{0} − x_{L}⟩ ≤ f_{L}(x_{0}) − f_{L}(x_{L}) → 0. Since ξ′_{L} → ξ′_{0} and x_{L} ⇀ x̄ (for some subsequence), we get that ⟨ξ′_{0}|x̄ − x_{0}⟩ = 0. Thus, we have x̄ ∈ T_{f}(x_{0}) ∩ U_{f(x_{0})} = T_{f}(x_{0}) ∩ L_{f(x_{0})} = {x_{0}}. This means that x_{0} is the only weak cluster point of the bounded sequence {x_{L}}.

Corollary 3.17. Let f_{0} be a bounded convex function with bounded level sets defined on the reflexive Banach space Y. Assume V in (2.4) is U_{α}, where α > f_{0}(x_{0}). If f_{0} is G−convex at x_{0} with respect to Z, then the inverse mapping (3.1) is G−continuous at (f_{0}, ξ′_{0}) for each ξ′_{0} ∈ ∂f_{0}(x_{0}).

Proof. This follows from the previous result and the fact that the weak topology on Y is described by sequences.

As a corollary to Lemma 3.7 we have the equivalence of G−convexity at a point with respect to the level set and the tangent set. The following lemma shows this carries over to F−convexity. This will simplify the proof of the strong analog of Theorem 3.16.

Lemma 3.18. The convex function f is F−convex at x_{0} with respect to T_{f}(x_{0}) if and only if it is F−convex at x_{0} with respect to L_{f(x_{0})}.

Proof. Assume f is not F−convex at x_{0} with respect to L_{f(x_{0})}. Then there exists α > 0 such that the modulus of convexity of f at x_{0} is zero at α, which by the monotonicity of this modulus, we can assume has been chosen small enough that f is Lipschitz on B_{2α}(x_{0}). Without loss of generality, we assume that there is a sequence {x_{L}} ⊂ B_{α}(x_{0}) ∩ L_{f(x_{0})} such that 0 = μ(α) = lim e(α; x_{0}, y_{L}), where αy_{L} = x_{L} − x_{0}. Now define the bounded convex (sublinear) function g on Y by g(x) = df(x_{0}; x−x_{0}) = max_{∂f(x_{0})}⟨ξ′|x−x_{0}⟩. Thus, we get that g(x_{L}) → 0^{−}. Choose x̂ ∈ S_{α}(x_{0}) so that g(x̂) < g(x_{L}) for L large enough. For such L we choose z_{L} (≠ x_{0}) such that z_{L}, x_{L} and x̂ are co-linear and such that g(z_{L}) = 0. This gives us the inequalities

\begin{matrix}\frac{g(x_{L}) − g(x̂)}{2α} ≤ & \frac{g(x_{L}) − g(x̂)}{∥x_{L} − x̂∥} \\ ≤ & \frac{g(z_{L}) − g(x_{L})}{∥z_{L} − x_{L}∥} \end{matrix}

for large L. Thus, we see that ∥z_{L}−x_{L}∥ → 0. Now we exploit the Lipschitz continuity of f to get |f(x_{L})−f(z_{L})| ≤ γ∥z_{L} −x_{L}∥. This means that {z_{L}} ∈ T_{f}(x_{0}) and ∥z_{L} −x_{0}∥ ≥ α/2 for large enough L. Thus, if we let ȳ_{L} = (z_{L} − x_{0})/∥z_{L} − x_{L}∥, then e(α/2; x_{0}, ȳ_{L}) → 0, so that f is not F−convex at x_{0} with respect to T_{f}(x_{0}). The converse of this is proved in a similar fashion.

We will need some notation. Fix x_{0} ∈ Y and ξ′_{0} ∈ ∂f(x_{0}). For each δ ∈ IR we define the hyperplane and the family thereof by

\begin{matrix}H_{δ} = H_{δ}(ξ′_{0}) := \{x ∈ Y : ⟨ξ′_{0}|x − x_{0}⟩ = −δ\} \\ H_{I} = H_{I}(ξ′_{0}) := \{x ∈ Y : x ∈ H_{δ} for some δ ∈ I\}. \end{matrix}

If I = [0, δ] for some δ > 0, we get sup{∥x − x_{0}∥ : x ∈ H_{I}(ξ′_{0}) ∩ L_{f(x_{0})}} ≤ μ̂^{−1}(δ) from inequality (3.4) for all δ > 0 and for all ξ′_{0} ∈ ∂f(x_{0}).

As we will be approximating a function f by a sequence of such functions {f_{L}}, we adopt the notation L_{α}^{L} := {x ∈ Y : f_{L}(x) = α} for the level sets of the approximating functions.

We let U := ∪_{β≤f(x_{0})}L_{β}, and similarly we let U^{L} := ∪_{β≤f_{L}(x_{L})}L_{β}^{L}. Finally, we use the notation H_{2I} := ∪_{δ≥μ̂(α)}H_{δ} and H_{I} := ∪_{δ≥μ̂(α)/2}H_{δ} to represent these half spaces.

We now consider a strong version of Theorem 3.16. We will see that F−convexity is a much stronger condition, allowing us to relax many of the conditions needed with G−convexity.

Theorem 3.19. Let f_{0} be a bounded convex function defined on a Banach space Y. Given a sequence of bounded convex functions {f_{L}} and a null sequence {δ_{L}} such that |f_{L}(x) − f_{0}(x)| < δ_{L} for all x ∈ B_{ρ}(x_{0}), where ρ > 0, assume that there is {x_{L}} with ξ′_{L} ∈ ∂f_{L}(x_{L}) such that ξ′_{L} → ξ′_{0} ∈ ∂f_{0}(x_{0}). If f_{L}(x_{L}) → f_{0}(x_{0}), say |f_{L}(x_{L}) − f_{0}(x_{0})| ≤ δ_{L}, and if f_{0} is F−convex at x_{0} with respect to Z, then x_{L} → x_{0}.

Proof. We first consider the case Z = L_{f_{0}(x_{0})}. Let α ∈ (0, ρ/2) be given. We show that there are only finitely many α_{L} := ∥x_{L} − x_{0}∥ ≥ 2α. Now F−convexity at x_{0} ensures that U ∩ S_{α}(x_{0}) ⊂ U ∩ H_{2I}. Uniform convergence assures us that there is some L_{1} ∈ IN such that U^{L} ∩ S_{α}(x_{0}) ⊂ U^{L} ∩ H_{I} for all L ≥ L_{1}. We extend this by the convexity of f_{L} to U^{L} ∩ (B_{α}(x_{0}))^{c} ⊂ U^{L} ∩ H_{I} for all L ≥ L_{1}.

Since 0 ∉ ∂f_{0}(x_{0}), we can find a x̂_{0} ∈ B_{α}(x_{0}) with f_{0}(x̂_{0}) < f_{0}(x_{0}). Uniform convergence of the sequence {f_{L}} to f_{0} guarantees that we can find a zero sequence {τ_{L}} and an L_{2} ≥ L_{1} such that x̄_{L} := τ_{L}(x̂_{0} − x_{0}) + x_{0} ∈ L_{f_{L}(x_{L})}^{L} ∩ B_{α}(x_{0}) for all L ≥ L_{2}.

For each L ∈ IN such that x_{L} ∉ {x_{0}, x̄_{L}} we define ȳ_{L} := (x_{L} − x̄_{L})/∥x_{L} − x̄_{L}∥ and y_{L} := (x_{L} − x_{0})/α_{L}. One can readily see that ∥ȳ_{L} − y_{L}∥ ≤ 2∥x̄_{L} − x_{0}∥/α_{L}.

Now consider the set of all L ≥ L_{2} for which λ_{L} := 2α/α_{L} ≤ 1, which we denote by J. We will show this set to be finite. By the convexity of each U^{L} we have λ(x_{L} −x̄_{L})+x̄_{L} ∈ U^{L} for any λ ∈ [0, 1]. We choose λ = λ_{L} in this expression and write

\begin{matrix}λ_{L}(x_{L} − x̄_{L}) + x̄_{L} = & λ_{L}(α_{L}y_{L} + x_{0} − x̄_{L}) + (x̄_{L} − x_{0}) + x_{0} \\ = & (αy_{L} + x_{0}) + [αy_{L} + (1 − λ_{L}) · (x̄_{L} − x_{0})]. \end{matrix}

Now the term in brackets is an element of B_{α}(αy_{L}) for all L ∈ J. Thus, for such L we get the set inclusion λ_{L}(x_{L} − x̄_{L}) + x̄_{L} ∈ U^{L} ∩ (B_{α}(x_{0}))^{c} ⊂ U^{L} ∩ H_{I}. If we note that 0 ≤ ⟨ξ′_{L}|x − x̄_{L}⟩ for all x ∈ L_{f_{L}(x_{L})}^{L} and all L ∈ IN, specifically for x_{L} ∈ L_{f_{L}(x_{L})}^{L}, we get

the inequality 0 ≤ ⟨ξ′_{L}|ȳ_{L}⟩, which lets us write

\begin{matrix}μ̂(α)/2 ≤ & − ⟨ξ′_{0}|λ_{L}(x_{L} − x̄_{L}) + x̄_{L} − x_{0}⟩ \\ ≤ & − ⟨ξ′_{0}|2αy_{L} + (1 − λ_{L})(x̄_{L} − x_{0})⟩ \\ ≤ & 2α [⟨ξ′_{L} − ξ′_{0}|ȳ_{L}⟩ + ⟨ξ′_{0}|ȳ_{L} − y_{L}⟩] + |⟨ξ′_{0}|x̄_{L} − x_{0}⟩| \\ ≤ & 2α∥ξ′_{L} − ξ′_{0}∥ + 3∥ξ′_{0}∥ · ∥x̂_{0} − x_{0}∥ · |τ_{L}| \end{matrix}

Since |τ_{L}| → 0 and ∥ξ′_{L} − ξ′_{0}∥ → 0 but μ̂(α) > 0, it follows that J is finite. Thus, we have {x_{L}} ∩ (B_{2α(x_{0})})^{c} is finite for each α ∈ (0, ρ/2), or in other words x_{L} → x_{0}.

Now consider the case Z = Y. Again, we let α_{L} = ∥x_{L} − x_{0}∥ and choose y_{L} ∈ S_{1} such that α_{L}y_{L} = x_{L} − x_{0}. Now define e_{L} in terms of f_{L} as in (2.2). Then if μ is the modulus of convexity, we can write

\begin{matrix}μ(α_{L}) ≤ & e_{0}(α_{L}; x_{0}, y_{L}) + e_{L}(α_{L}; x_{0}, −y_{L}) \\ = & \frac{f_{0}(x_{L}) − f_{0}(x_{0}) − f_{L}(x_{L}) + f_{L}(x_{0})}{α_{L}} − df_{L}(x_{L}; −y_{L}) − df_{0}(x_{0}; y_{L}) \\ ≤ & \frac{2δ_{L}}{α_{L}} + ∥ξ′_{L} − ξ′_{0}∥ \end{matrix}

This of course implies that x_{L} → x_{0}.

It should be noted that we have shown that for any ε > 0 there is an L_{ε} such that if L ≥ L_{ε}, then ∥x_{L} − x_{0}∥ < ε and this L_{ε} is independent of ξ′_{0} ∈ ∂f(x_{0}). Under suitable conditions this theorem can be made uniform in x ∈ V. Similar in spirit and proof we get

Corollary 3.20. Let f_{0} be a bounded convex function defined on the Banach space Y. Assume V in (2.4) is B_{ρ}(x_{0}), where ρ > 0. If f_{0} is F−convex at x_{0} with respect to Z, then the inverse mapping (3.1) is uniformly F−continuous on {f_{0}} × ∂f_{0}(x_{0}).

In the next section we will invert this last result in the case of norms in the process of deriving a local Smulyan theorem. That is, we will show that the bornological approach to smoothness and convexity allows for the characterization locally of one of these properties in the primal space by the dual property in the dual space.

4. Applications

We begin this section by giving a sufficient condition for F−convexity at a point x_{0}. This condition reminds us of the definition of uniform convexity in [2] and [5] for the case of the norm on Y. We go on to consider the duality between differentiability and convexity.

4.1. Local Uniform Rotundity

Once again the Mean Value Theorem provides us with the machinery to obtain

Theorem 4.1. If f is uniformly rotund at x_{0} with respect to Z, that is if there exists a modulus function δ ∈ C_{0}(IR) such that for all x ∈ Z we have

f (\frac{x + x_{0}}{2} ) ≤ \frac{f(x) + f(x_{0})}{2} − δ (∥x − x_{0}∥) ,  (4.1)

then f is F−convex at x_{0} with respect to Z.

Proof. We consider inequality (4.1), where we assume that x ≠ x_{0}. Define the modulus function μ_{x_{0}} : [0, ∞) → [0, ∞) by

μ_{x_{0}}(α) := \{ \begin{matrix}2δ(α)/α & if α > 0, \\ 0 & if α = 0. \end{matrix}

Since δ(α) > 0 for all α > 0, we get that μ_{x_{0}}(α) > 0 for all α > 0. We will simplify our notation by defining α := ∥x − x_{0}∥ > 0 and y := (x − x_{0})/α.

If Z = Y, we rearrange (4.1) and use the results of Section 2.2 to get the inequality

\begin{matrix}μ_{x_{0}}(α) ≤ & \frac{f(x_{0} + αy) − f(x_{0})}{α} − \frac{f(x_{0} + (α/2) · y) − f(x_{0})}{α/2} \\ ≤ & \frac{f(x_{0} + αy) − f(x_{0})}{α} − ⟨ξ′|y⟩, \end{matrix}

which is valid for all α > 0, for all ξ′ ∈ ∂f(x_{0}) and for all y ∈ S_{1}.

If Z = L_{f(x_{0})}, let x ∈ Z ∩ S_{α}(x_{0}). From (2.1) and (4.1) we see that for all ξ′ ∈ ∂f(x_{0}), we can derive the inequality

\begin{matrix}⟨ξ′ | \frac{x − x_{0}}{2} ⟩ ≤ & f (\frac{x − x_{0}}{2} + x_{0}) − f(x_{0}) \\ ≤ & − δ (∥x − x_{0}∥) . \end{matrix}

Dividing this by α/2 and rearranging gives us μ_{x_{0}}(α) ≤ −⟨ξ′|y⟩ for all ξ′ ∈ ∂f(x_{0}) and all y ∈ S_{1} with x_{0} + αy ∈ L_{f(x_{0})}. If dia(L_{f(x_{0})}) ≠ ∞, we must redefine μ_{x_{0}} a bit but it is obvious how to do this. The proof for Z = T_{f}(x_{0}) is similar to this proof.

We note that Theorem 4.1 and Lemma 3.9 together show the modulus function δ is necessarily o(α) as α → 0^{+}. For a different proof of this see [6].

We also note that (4.1) is by no means necessary for the F−convexity at x_{0} ∈ Y as we will see in the next example. We motivate the claim that local uniform rotundity as developed in [10] does not sufficiently capture the local duality at play between smoothness and convexity of dual norms. It is designed to measure the convexity of the norm at a point x_{0} ∈ S_{1} in terms of points at some distance from x_{0}. Thus, the smoothness of the norm on a neighborhood of x_{0} can play a role in measuring its convexity, and clearly for a satisfactory duality theory, differentiability should play no such role in the primal space (only in the dual space). In our definition, however, we examine the subdifferential of f at x_{0}, and thereby avoid the unfortunate situation of involving the smoothness of f on a neighborhood of x_{0} in the measure of the convexity of f at x_{0}.

Example 4.2. We will introduce an equivalent norm f on ℓ^{2} which is F−convex at e_{1} but not uniformly rotund there. We start by defining the family of subsets of ℓ^{2}

Z_{j} := { x := (x_{i})_{i∈IN} ∈ ℓ^{2} : ( |x_{1}|^{(j+1)/j} + |x_{j}|^{(j+1)/j} ) ≤ 1} ∩ B̄_{1}

for each j ≥ 2. Then we define Z := ∩_{j≥2}Z_{j}. It is not difficult to see that the set inclusion

B̄\sqrt{1/2} ⊂ Z ⊂ B̄_{1}  (4.2)

is valid. Next we define f on ℓ^{2} by f(x) := inf{α > 0 : x/α ∈ Z}, which, due to (4.2), induces a norm on ℓ^{2} equivalent to that derived from the scalar product. If we define x_{L} := (e_{1} + e_{L})/2 ∈ ℓ^{2} for each L ≥ 2, then it is easy to see that f(x_{L}) = 2^{−1/(L+1)} → 1.

Since f(e_{j}) = 1 for all j ∈ IN, we get that f is not uniform rotund at e_{1}. Now let ξ′ ∈ ∂f(e_{1}). For all j ∈ IN and for all β ∈ (0, 1) the inequalities

\frac{1 − f(e_{1} − βe_{j})}{β} ≤ ⟨ξ′|e_{j}⟩ ≤ \frac{f(e_{1} + βe_{j}) − 1}{β}

are valid. Since f(e_{1} ± βe_{j}) = (1 + β^{(j+1)/j})^{j/(j+1)} for j ≥ 2 and f(e_{1} ± βe_{1}) = 1 ± β, we can use L’Hospital’s rule and the above inequalities to get ⟨ξ′|e_{j}⟩ = δ_{1,j} (Kronecker’s δ), so for all ξ′ ∈ ∂f(e_{1}) we get ⟨ξ′|y⟩ = y_{1}. We want to check the F−convexity at e_{1}, so we consider any y ∈ S_{1} with e_{1} + αy ∈ L_{1}. Since L_{1} ⊂ B̄_{1}, we have y_{1} < 0, and we get some β ≥ 1 such that e_{1} + βαy ∈ S_{1}. Now, one can easily check that the norm on any Hilbert space is F−convex at each x_{0} ∈ S_{1} with respect to the unit sphere, and for such a norm the modulus of convexity μ_{H} is exactly μ_{H}(α) = α/2. Putting the last few ideas together gives

α/2 = μ_{ℓ^{2}}(α) ≤ μ_{ℓ^{2}}(βα) ≤ −⟨e_{1}, y⟩ = |y_{1}| = −⟨ξ′|y⟩

for all ξ′ ∈ ∂f(e_{1}) and for all y ∈ S_{1} such that e_{1} + αy ∈ L_{1}. Thus, we get that f is F−convex at e_{1}, but not uniformly rotund there.

It is interesting to note that if f is uniformly rotund at x_{0} and F^{+}−differentiable there, then f is also F−convex at x_{0}.

Of course, one of the nice features of a uniformly convex Banach space, that is a Banach space whose norm satisfies (4.1) for all x, x_{0} ∈ S_{1}, is that if {x_{L}} ⊂ S_{1} and x_{L} ⇀ x_{0} ∈ S_{1}, then x_{L} → x_{0}. Theorem 4.1 gives us an elementary proof of this fact.

Corollary 4.3. Let f be a norm, which is uniformly rotund at x_{0} ∈ S_{1}, locally uniformly convex in the sense of [10]. If {x_{L}} ⊂ S_{1} is any sequence such that x_{L} ⇀ x_{0}, then x_{L} → x_{0}.

Proof. Since uniform rotundity of f at x_{0} implies the F−convexity of f there, we can define μ as in Theorem 4.1 to get ∥x_{L} − x_{0}∥ · μ(∥x_{L} − x_{0}∥) ≤ |⟨ξ′|x_{L} − x_{0}⟩|. This and x_{L} ⇀ x_{0} ∈ S_{1} give the norm convergence we want.

4.2. Applications to Banach Space Geometry

We are now ready to consider more closely the special case where f is the norm on the Banach space Y. We will use both f and ∥·∥ to denote this norm (the notation f reminds us that the norm is a convex function, and ∥ · ∥ measures the distance between points). Likewise, we will use f^{∗} and ∥ · ∥ to denote the norm on Y′, the dual of Y. We assume V ⊆ S_{1} and define V′ := ∪_{x∈V}∂f(x) ⊆ S′_{1}. We remind the reader of some of the notation of Sections 2 and 3 that we will employ. In this notation

ξ′ ↦ ∂f^{∗}(ξ′) is exactly the subdifferential mapping from S′_{1} to S′′_{1}, and
ξ′ ↦ ∂f_{S_{1}}^{∗}(ξ′) is the restriction of this of this mapping to V′ (with range S_{1}).

We will need the following generalization of a lemma which can be found in [5].

Lemma 4.4. Let f be a norm on Y, which is uniformly F−differentiable on V ⊆ S_{1}. For every ε > 0 there exists a δ > 0 such that if ξ′ ∈ S′_{1} and ⟨∇f(x) − ξ′|x⟩ < δ, then ∥∇f(x) − ξ′∥ < ε. Further, this δ is independent of x ∈ V.

Proof. Assume ∥ξ′ −∇f(x)∥ = ε. Choose y ∈ S_{1} such that ⟨ξ′ −∇f(x)|y⟩ ≥ ε/2. Define α := 4⟨∇f(x) − ξ′|x⟩/ε > 0. Since ξ′ ∉ ∂f(x) = {∇f(x)}, α is positive. Then we have

\begin{matrix}⟨∇f(x) − ξ′|x⟩ ≤ & ⟨∇f(x) − ξ′|x⟩[4⟨ξ′ − ∇f(x)|y⟩/ε − 1] \\ = & ⟨ξ′ − ∇f(x)|x⟩ + α⟨ξ′ − ∇f(x)|y⟩ \\ = & ⟨ξ′|x + αy⟩ − ∥x∥ − α⟨∇f(x)|y⟩ \\ ≤ & ∥x + αy∥ − ∥x∥ − α⟨∇f(x)|y⟩ \\ ≤ & αE(α). \end{matrix}

Thus, we derive the inequality E^{−1}(ε/4) ≤ α, which gives us the inequality

εE^{−1}(ε/4)/4 ≤ ⟨∇f(x) − ξ′|x⟩.  (4.3)

If we define the function Ê ∈ C_{0}(IR) implictly by Ê^{−1}(α) := [αE^{−1}(α)] for all α > 0, then inequality (4.3) yields ∥ξ′ − ∇f(x)∥ ≤ 4Ê( ⟨∇f(x) − ξ′|x⟩ ), which proves our claim.

We start by giving a second sufficient condition for G − [F−]convexity for the case of the norm on a Banach space. This is a local version of Smulyan’s theorem, which states that a Banach space with a uniformly Fréchet differentiable dual norm is a uniformly convex space, that is its norm satisfies (4.1) at each x_{0} ∈ S_{1} with a uniform modulus δ.

Theorem 4.5. In the above notation let f^{∗} be G-differentiable on V′ [uniformly Fdifferentiable on V′]. Then f is G − [uniformly F−]convex on V with respect to S_{1}.

Proof. The G-differentability of f^{∗} on V′, Theorem 2.7 and the isometric embedding of Y in Y′′ give the G− (or norm to weak) continuity of ξ′ ↦ ∂f_{S_{1}}^{∗}(ξ′) on V′, so that the results follow by a direct application of Theorem 3.12.

If we assume uniform F-differentiability on V′, then we can derive μ directly from the upper estimate E that we can get for e in (2.2). To these ends we embed S_{1} in S′′_{1} and use Lemma 4.4 applied to Y′. Choose α ∈ (0, 2), choose x ∈ V, choose ξ′ ∈ ∂f(x) and choose y ∈ S_{1} such that x + αy ∈ S_{1}. By the F-differentiability of f^{∗} at ξ′ we have x = ∇f^{∗}(ξ′), so that (4.3) together with the pairing ⟨ξ′|y⟩ = ⟨y|ξ′⟩ gives us

αE^{−1}(α/4)/4 ≤ ⟨∇f^{∗}(ξ′) − (x + αy)|ξ′⟩
= − α⟨ξ′|y⟩,  (4.4)

where E ∈ C_{0}(IR) is the uniform estimate ensured by the smoothness of f^{∗}. Define μ : [0, ∞) → IR by μ(α) := E^{−1}(α/4)/4 > 0 for all α > 0. Thus, inequality (4.4) shows that μ(α) ≤ inf_{β≥α}{e(β; x, y) : y ∈ S_{1} ∩ Z_{β}(x)} for all x ∈ V. This means that by definition f is uniformly F−convex on V with respect to the unit sphere.

We have shown not only the implications that the smoothness of the dual norm has for the convexity of the primal norm, but we also found the modulus of convexity of the primal norm in terms of the estimate on the smoothness of the dual norm. We will turn this around shortly and show how to derive an estimate of the smoothness of the primal norm in terms of the modulus of convexity of the dual norm. Our first result reiterates Corollary 3.20, while adding uniformity in x ∈ V. It is actually valid for any bounded, convex function.

Lemma 4.6. Let f be uniformly F−convex on V ⊆ S_{1}. Then ξ′ ↦ ∂f_{S_{1}}^{∗}(ξ′) is norm to norm uniformly continuous on V′ := ∪_{x∈V}∂f(x).

Proof. Let x ∈ S_{α}(x_{0}) for some α > 0 and define y := (x − x_{0})/α. Using the moduli of convexity of f on V and at x we can write μ_{x}(α) ≤ ⟨ξ′|y⟩ for any ξ′ ∈ ∂f(x), and we can write μ_{V}(α) ≤ −⟨ξ′_{0}|y⟩ for any ξ′_{0} ∈ ∂f(x_{0}). This gives

μ_{V}(α) ≤ μ_{V}(α) + μ_{x}(α)
≤ ⟨ξ′ − ξ′_{0}|y⟩  (4.5)
≤ ∥ξ′ − ξ′_{0}∥.

Now since μ_{V}(α) > 0 if α > 0 and continuous at the origin, we can invert it (taking the obvious precautions if μ_{V} is constant or discontiuous anywhere), so that from (4.5) we get

∥x − x_{0}∥ ≤ μ_{V}^{−1}(∥ξ′ − ξ′_{0}∥)  (4.6)

for any x ∈ S_{α}(x_{0}), for any ξ′_{0} ∈ ∂f(x_{0}) and for any ξ′ ∈ ∂f(x). Since μ_{V} is the modulus of convexity for all x_{0} ∈ V, we get norm to norm uniform continuity that we seek.

4.3. Local Duality and Smulyan’s Theorem

Using the notation of Section 4.2, we can summarize our results. We begin with the weak equivalencies (for a reflexive space).

Theorem 4.7. Consider the statements:

(i) f is G−convex on V with respect to S_{1}. (ii) ξ′ ↦ ∂f_{S_{1}}^{∗}(ξ′) is single valued on V′. (iii) ξ′ ↦ ∂f^{∗}(ξ′) is single valued on V′. (iv) f^{∗} is G−differentiable on V′.

(v) ξ′ ↦ ∂f^{∗}(ξ′) is norm to weak^{∗} continuous on V′. (vi) ξ′ ↦ ∂f_{S_{1}}^{∗}(ξ′) is norm to weak continuous on V′.

Then we have (i)⇔(ii)⇐(iii)⇔(iv)⇔(v)⇔(vi), and all are equivalent if Y is reflexive.

Proof. In Theorem 3.10 we demonstrated (i)⇔(ii) pointwise for any bounded, convex function. The implication (ii)⇐(iii) is trivial, and we get equivalency if Y is reflexive. In [11] the author shows that (iii)⇔(iv) pointwise for any convex function. Theorem 2.7 gives (iv)⇔(v) pointwise. The implication (v)⇒(vi) is trivial, since the weak^{∗} topology on S′′_{1} ∩ S_{1} is the same as the weak topology on S_{1}. We get the equivalency by an argument similar to that showing (ii)⇒(iii) in the next result. Theorem 3.12 gives (vi)⇒(i) pointwise for any bounded, convex function with equivalency if Y is reflexive.

Theorem 4.8. (A local Smulyan result) The following are equivalent:

(i) f is uniformly F−convex on V with respect to S_{1}.

(ii) ξ′ ↦ ∂f_{S_{1}}^{∗}(ξ′) is norm to norm uniformly continuous on V′. (iii) ξ′ ↦ ∂f^{∗}(ξ′) is norm to norm uniformly continuous on V′. (iv) f^{∗} is uniformly F−differentiable on V′.

Proof. The implication (i)⇒(ii) follows from Lemma 4.6. By arguments similar to those in [5], or by a modification of Theorem 2.9 we get (iii)⇒(iv). Theorem 4.5 gives us the implication (iv)⇒(i). That (ii)⇒(iii) can be derived by applying monotonicity of the subdifferential mapping and Bishop-Phelps in the following way. Consider some δ−ball B_{δ}(ξ′_{0}) about ξ′_{0} ∈ S′_{1}. We let Ŷ be the embedding of Y in Y′′ and let X′′ be the weak^{∗} closure of the convex hull of the set ∂f^{∗}(B_{δ}(ξ′_{0})) ∩ Ŷ. Suppose there is η′′_{0} ∈ ∂f^{∗}(B_{δ}(ξ′_{0}))\X′′. Choose ξ′ ∈ S′_{1} such that ⟨η′′_{0}|ξ′⟩ > α > max_{X′′}⟨η′′|ξ′⟩ for some α > 0. Clearly, we can find a η′ ∈ B_{δ}(ξ′_{0}) such that η′′ ∈ ∂f^{∗}(η′). Choose β > 0 small enough that η′ + βξ′ ∈ B_{δ}(ξ′_{0}). Monotonicity of the subdifferential operator gives

\begin{matrix}α < & ⟨η′′|ξ′⟩ \\ ≤ & max_{∂f^{∗}(η′)}⟨ω′′|ξ′⟩ \\ ≤ & min_{∂f^{∗}(η′+βξ′)}⟨ω′′|ξ′⟩. \end{matrix}

By weak^{∗} upper semi-continuity (see Theorem 2.7 or [11]) of the subdifferential mapping, there exists an open neighborhood V of η′ + βξ′ such that inf_{∂f^{∗}(V)}⟨ω′′|ξ′⟩ > α. But this contradicts the Bishop-Phelps theorem which says that ∂f^{∗}(V) ∩ X′′ ≠ ∅. Thus, we have ∂f^{∗}(B_{δ}(ξ′_{0})) ⊆ X′′. Now, (ii) means that for every ε > 0 there is some δ > 0 such that if ξ′ ∈ V′ and x ∈ ∂f(ξ′).then we get the set inclusion

∂f_{S_{1}}^{∗}(B_{δ}(ξ′)) ⊆ B_{ε}(x).

Now we take the weak^{∗} closure of the convex hull of the above inclusion, and apply the results we just derived and we get exactly (iii). Note that the same ε and δ hold [7].

Next we show the impact that convexity of the dual norm has on the smoothness of the primal norm. This is the dual concept of Theorem 4.5 and allows us to derive a modulus of smoothness for the primal norm in terms of the modulus of convexity of the dual norm. Theorem 4.9. If f^{∗} is G− [uniformly F−] convex on V′ with respect to the dual unit sphere, then f is G- [uniformly F-]differentiable on V. For G−convexity we need reflexivity.

Proof. Reflexivity of Y and the G−convexity of f^{∗} on V′ allows us to apply Theorem 4.7 to get this result. Likewise, we can apply Theorem 4.8 to get that the second dual norm is uniformly F−differentiable on the restriction of V′′ back to V ⊆ S_{1}, giving the result. We now go on to derive a modulus of smoothness. Assuming uniform convexity we can apply (4.6) to the proof of Theorem 2.7 to get

\begin{matrix}αe(α; x_{0}, y) = & f(x_{0} + αy) − f(x_{0}) − αdf(x_{0}); y) \\ ≤ & α⟨ξ′_{α} − ξ′_{0}|y⟩ \\ ≤ & |α| · ∥ξ′_{α} − ξ′_{0}∥ \\ ≤ & |α| · μ_{V}^{−1}(|α|), \end{matrix}

J. Read / A bornological approach to rotundity and smoothness 139

where ξ′_{α} ∈ ∂f(x_{0} + αy) and ξ′_{0} ∈ ∂f(x_{0}). Define the function Ê on the set (−δ, δ) by Ê(α) := sgn(α)μ_{W}^{−1}(|α|) for some δ > 0. Then Ê is continuous at the origin and satisfies |Ê(α)| > 0 if α > 0. Thus, we can find some E ∈ C_{0}(IR) with αÊ(α) ≤ αE(α) for all α ∈ [−δ, δ] and we get uniform F-differentiability of f on V.

4.4. Applications to Convergence Rates The last section puts us in a position to get a convergence rate for Theorem 3.19 in the case of the norm as follows. Theorem 4.10. Assume that f is uniformly F−differentiable on B_{ρ}(x_{0}) for some x_{0} ∈ S_{1} and some ρ ∈ (0, 1) and F−convex with respect to S_{1} at x_{0} with modulus μ. Let f_{L} be a sequence of norms on Y such that |f_{L}(x) − f(x)| < δ_{L} → 0 for all x ∈ B_{ρ}(x_{0}). Further, let {x_{L}} ⊂ Y with f_{L}(x_{L}) → 1 and ξ′_{L} → ∇f(x_{0}) (where ξ′_{L} ∈ ∂f_{L}(x_{L})), then we can derive a convergence rate in terms of f(x_{L}), δ_{L} and ∥ξ′_{L} − ∇f(x_{0})∥.

Proof. By Theorem 3.19 there is some L_{0} ∈ IN such that x_{L} ∈ B_{ρ}(x_{0}) for L ≥ L_{0}. For any such L we can define x̂_{L} := x_{L}/f(x_{L}) ∈ S_{1}. The results of Theorem 2.9 allow us to write ∥ξ′_{L} − ∇f(x)∥ ≤ 2\sqrt{δ_{L}} + E(\sqrt{δ_{L}}) for all x ∈ B_{ρ}(x_{0}). This estimate together with (4.6) and the positive homogeneity of f give

\begin{matrix}∥x_{L} − x_{0}∥ ≤ & ∥x_{L} − x̂_{L}∥ + ∥x̂_{L} − x_{0}∥ \\ ≤ & |f(x_{L}) − 1| + μ^{−1}(∥∇f(x_{L}) − ∇f(x_{0})∥) \\ ≤ & |f(x_{L}) − 1| + μ^{−1}(∥∇f(x_{L}) − ξ′_{L}∥ + ∥ξ′_{L} − ∇f(x_{0})∥) \\ ≤ & |f(x_{L}) − 1| + μ^{−1}(2\sqrt{δ_{L}} + E(\sqrt{δ_{L}}) + ∥ξ′_{L} − ∇f(x_{0})∥), \end{matrix}

which is what we want to show.

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