Hyperplane Sections of Convex Bodies in Isotropic Position

Abstract: We prove sharp inequalities showing that the volume of the hyperplane sections of a convex body in isotropic position through its centroid, as well as those of maximal volume in a fixed direction, does not depend much of the hyperplane. This generalizes a result of K. Ball to non-symmetric convex bodies. For the sections through its centroid, the extremal bodies are the same as in the symmetric case (cylinders and double-cones). For the sections of maximal volume parallel to a fixed hyperplane, the extremal bodies are cylinders and cones. We deduce that the cross-section body and the intersection body of a convex body with respect to its centroid are close to its Binet ellipsoid; we generalize results of J. Bourgain on maximal functions to the case of non-symmetric convex bodies and we prove that some equivalences between different forms of the hyperplane conjecture still hold in the non-symmetric case.

Keywords: volume, section, centroid, isotropic, ellipsoid, inertia, hyperplane, intersection body, cross-section body, maximal function

Classification (MSC91): 52A20, 52A40

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