@article {colorgale,
	AUTHOR = {Adiprasito, Karim A. and Brinkmann, Philip and Padrol, Arnau
	and Pat\'{a}k, Pavel and Pat\'{a}kov\'{a}, Zuzana and Sanyal, Raman},
	TITLE = {Colorful simplicial depth, {M}inkowski sums, and generalized
	{G}ale transforms},
	JOURNAL = {Int. Math. Res. Not. IMRN},
	FJOURNAL = {International Mathematics Research Notices. IMRN},
	YEAR = {2019},
	NUMBER = {6},
	PAGES = {1894--1919},
	ISSN = {1073-7928},
	MRCLASS = {52A35 (05A18 05E45 52B05 52B35)},
	MRNUMBER = {3932598},
	MRREVIEWER = {P. McMullen},
	DOI = {10.1093/imrn/rnx184},
	URL = {https://doi.org/10.1093/imrn/rnx184},
}



@inproceedings {edmonds,
    AUTHOR = {Edmonds, Jack},
     TITLE = {Submodular functions, matroids, and certain polyhedra},
 BOOKTITLE = {Combinatorial {S}tructures and their {A}pplications ({P}roc.
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     PAGES = {69--87},
 PUBLISHER = {Gordon and Breach, New York},
      YEAR = {1970},
   MRCLASS = {05.35},
  MRNUMBER = {0270945},
MRREVIEWER = {A. W. Ingleton},
}
		

@article {meyer,
	AUTHOR = {Meyer, Walter},
	TITLE = {Indecomposable polytopes},
	JOURNAL = {Trans. Amer. Math. Soc.},
	FJOURNAL = {Transactions of the American Mathematical Society},
	VOLUME = {190},
	YEAR = {1974},
	PAGES = {77--86},
	ISSN = {0002-9947},
	MRCLASS = {52A25},
	MRNUMBER = {338929},
	MRREVIEWER = {P. McMullen},
	DOI = {10.2307/1996951},
	URL = {https://doi.org/10.2307/1996951},
}

@article {genasso,
    AUTHOR = {Hohlweg, Christophe and Lange, Carsten E. M. C. and Thomas,
              Hugh},
     TITLE = {Permutahedra and generalized associahedra},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {226},
      YEAR = {2011},
    NUMBER = {1},
     PAGES = {608--640},
      ISSN = {0001-8708},
   MRCLASS = {20F55 (05E15 52B11)},
  MRNUMBER = {2735770},
MRREVIEWER = {Chi Kin Mak},
       DOI = {10.1016/j.aim.2010.07.005},
       URL = {https://doi.org/10.1016/j.aim.2010.07.005},
}

@article {gentle,
    AUTHOR = {Palu, Yann and Pilaud, Vincent and Plamondon, Pierre-Guy},
     TITLE = {Non-kissing complexes for gentle algebras},
   JOURNAL = {S\'{e}m. Lothar. Combin.},
  FJOURNAL = {S\'{e}minaire Lotharingien de Combinatoire},
    VOLUME = {80B},
      YEAR = {2018},
     PAGES = {Art. 39, 12},
   MRCLASS = {05E45},
  MRNUMBER = {3940614},
}

@book {grunb,
	AUTHOR = {Gr\"{u}nbaum, Branko},
	TITLE = {Convex polytopes},
	SERIES = {Graduate Texts in Mathematics},
	VOLUME = {221},
	EDITION = {Second edition},
	NOTE = {Prepared and with a preface by Volker Kaibel, Victor Klee and
	G\"{u}nter M. Ziegler},
	PUBLISHER = {Springer-Verlag, New York},
	YEAR = {2003},
	PAGES = {xvi+468},
	ISBN = {0-387-00424-6; 0-387-40409-0},
	MRCLASS = {52-01 (52Bxx)},
	MRNUMBER = {1976856},
	MRREVIEWER = {Alexander Zvonkin},
	DOI = {10.1007/978-1-4613-0019-9},
	URL = {https://doi.org/10.1007/978-1-4613-0019-9},
}

@book {realizations,
    AUTHOR = {Richter-Gebert, J\"{u}rgen},
     TITLE = {Realization spaces of polytopes},
    SERIES = {Lecture Notes in Mathematics},
    VOLUME = {1643},
 PUBLISHER = {Springer-Verlag, Berlin},
      YEAR = {1996},
     PAGES = {xii+187},
      ISBN = {3-540-62084-2},
   MRCLASS = {52B05 (14P10 52B55)},
  MRNUMBER = {1482230},
MRREVIEWER = {Margaret M. Bayer},
       DOI = {10.1007/BFb0093761},
       URL = {https://doi.org/10.1007/BFb0093761},
}

@inproceedings {klee-minty,
	AUTHOR = {Klee, Victor and Minty, George J.},
	TITLE = {How good is the simplex algorithm?},
	BOOKTITLE = {Inequalities, {III} ({P}roc. {T}hird {S}ympos., {U}niv.
	{C}alifornia, {L}os {A}ngeles, {C}alif., 1969; dedicated to
	the memory of {T}heodore {S}. {M}otzkin)},
	PAGES = {159--175},
	YEAR = {1972},
	MRCLASS = {90C05},
	MRNUMBER = {0332165},
	MRREVIEWER = {Robert G. Jeroslow},
}

@article {weights,
	AUTHOR = {McMullen, P.},
	TITLE = {Weights on polytopes},
	JOURNAL = {Discrete Comput. Geom.},
	FJOURNAL = {Discrete \& Computational Geometry. An International Journal
	of Mathematics and Computer Science},
	VOLUME = {15},
	YEAR = {1996},
	NUMBER = {4},
	PAGES = {363--388},
	ISSN = {0179-5376},
	MRCLASS = {52B05},
	MRNUMBER = {1384882},
	MRREVIEWER = {Tiong Seng Tay},
	DOI = {10.1007/BF02711515},
	URL = {https://doi.org/10.1007/BF02711515},
}


@article {faces,
	AUTHOR = {Postnikov, Alex and Reiner, Victor and Williams, Lauren},
	TITLE = {Faces of generalized permutohedra},
	JOURNAL = {Doc. Math.},
	FJOURNAL = {Documenta Mathematica},
	VOLUME = {13},
	YEAR = {2008},
	PAGES = {207--273},
	ISSN = {1431-0635},
	MRCLASS = {05E30 (05A15)},
	MRNUMBER = {2520477},
}

@article {eisenbud,
	AUTHOR = {Eisenbud, David and Popescu, Sorin},
	TITLE = {The projective geometry of the {G}ale transform},
	JOURNAL = {J. Algebra},
	FJOURNAL = {Journal of Algebra},
	VOLUME = {230},
	YEAR = {2000},
	NUMBER = {1},
	PAGES = {127--173},
	ISSN = {0021-8693},
	MRCLASS = {14N05 (14M05 14N15)},
	MRNUMBER = {1774761},
	MRREVIEWER = {I. Dolgachev},
	DOI = {10.1006/jabr.1999.7940},
	URL = {https://doi.org/10.1006/jabr.1999.7940},
}

@misc{karim,
	title={On the realization space of the cube},
	author={Karim Adiprasito and Daniel Kalmanovich and Eran Nevo},
	year={2019},
	eprint={1912.09554},
	archivePrefix={arXiv},
	primaryClass={math.CO}
}
@article {indecomposable,
	AUTHOR = {McMullen, Peter},
	TITLE = {Indecomposable convex polytopes},
	JOURNAL = {Israel J. Math.},
	FJOURNAL = {Israel Journal of Mathematics},
	VOLUME = {58},
	YEAR = {1987},
	NUMBER = {3},
	PAGES = {321--323},
	ISSN = {0021-2172},
	MRCLASS = {52A25},
	MRNUMBER = {917362},
	MRREVIEWER = {Juan Garc\'{\i}a Laguna},
	DOI = {10.1007/BF02771695},
	URL = {https://doi.org/10.1007/BF02771695},
}

@inproceedings {greek,
	AUTHOR = {Emiris, Ioannis Z. and Konaxis, Christos and Zafeirakopoulos,
	Zafeirakis},
	TITLE = {Minkowski decomposition and geometric predicates in sparse
	implicitization},
	BOOKTITLE = {I{SSAC}'15---{P}roceedings of the 2015 {ACM} {I}nternational
	{S}ymposium on {S}ymbolic and {A}lgebraic {C}omputation},
	PAGES = {157--164},
	PUBLISHER = {ACM, New York},
	YEAR = {2015},
	MRCLASS = {68W30 (52B20 65D17 68U07)},
	MRNUMBER = {3388295},
	MRREVIEWER = {G\'{a}bor Hetyei},
}

@article {mcmullen,
	AUTHOR = {McMullen, P.},
	TITLE = {Representations of polytopes and polyhedral sets},
	JOURNAL = {Geometriae Dedicata},
	FJOURNAL = {Geometriae Dedicata},
	VOLUME = {2},
	YEAR = {1973},
	PAGES = {83--99},
	ISSN = {0046-5755},
	MRCLASS = {52A25},
	MRNUMBER = {0326574},
	MRREVIEWER = {G. C. Shephard},
	DOI = {10.1007/BF00149284},
	URL = {https://doi.org/10.1007/BF00149284},
}


@book {triangulations,
	AUTHOR = {De Loera, Jes\'{u}s A. and Rambau, J\"{o}rg and Santos, Francisco},
	TITLE = {Triangulations},
	SERIES = {Algorithms and Computation in Mathematics},
	VOLUME = {25},
	NOTE = {Structures for algorithms and applications},
	PUBLISHER = {Springer-Verlag, Berlin},
	YEAR = {2010},
	PAGES = {xiv+535},
	ISBN = {978-3-642-12970-4},
	MRCLASS = {52B55 (05C10 52B05 57Q15 68U05)},
	MRNUMBER = {2743368},
	DOI = {10.1007/978-3-642-12971-1},
	URL = {https://doi.org/10.1007/978-3-642-12971-1},
}

@book {bible,
	AUTHOR = {McMullen, P. and Shephard, G. C.},
	TITLE = {Convex polytopes and the upper bound conjecture},
	NOTE = {Prepared in collaboration with J. E. Reeve and A. A. Ball,
	London Mathematical Society Lecture Note Series, 3},
	PUBLISHER = {Cambridge University Press, London-New York},
	YEAR = {1971},
	PAGES = {iv+184},
	MRCLASS = {52A25},
	MRNUMBER = {0301635},
	MRREVIEWER = {H. S. M. Coxeter},
}

@misc{cubes,
	title={On the realization space of the cube},
	author={Karim Adiprasito and Daniel Kalmanovich and Eran Nevo},
	year={2019},
	eprint={1912.09554},
	archivePrefix={arXiv},
	primaryClass={math.CO}
}

@article {coxeter,
	AUTHOR = {Ardila, Federico and Castillo, Federico and Eur, Christopher
	and Postnikov, Alexander},
	TITLE = {Coxeter submodular functions and deformations of {C}oxeter
	permutahedra},
	JOURNAL = {Adv. Math.},
	FJOURNAL = {Advances in Mathematics},
	VOLUME = {365},
	YEAR = {2020},
	PAGES = {107039},
	ISSN = {0001-8708},
	MRCLASS = {14M25 (05E 52B12)},
	MRNUMBER = {4064768},
	DOI = {10.1016/j.aim.2020.107039},
	URL = {https://doi.org/10.1016/j.aim.2020.107039},
}



@incollection {oriented,
	AUTHOR = {Richter-Gebert, J\"{u}rgen and Ziegler, G\"{u}nter M.},
	TITLE = {Oriented matroids},
	BOOKTITLE = {Handbook of discrete and computational geometry},
	SERIES = {CRC Press Ser. Discrete Math. Appl.},
	PAGES = {111--132},
	PUBLISHER = {CRC, Boca Raton, FL},
	YEAR = {1997},
	MRCLASS = {05B35 (52B40 90C57)},
	MRNUMBER = {1730162},
}

@book {ziegler,
	AUTHOR = {Ziegler, G\"{u}nter M.},
	TITLE = {Lectures on polytopes},
	SERIES = {Graduate Texts in Mathematics},
	VOLUME = {152},
	PUBLISHER = {Springer-Verlag, New York},
	YEAR = {1995},
	PAGES = {x+370},
	ISBN = {0-387-94365-X},
	MRCLASS = {52Bxx},
	MRNUMBER = {1311028},
	MRREVIEWER = {Margaret M. Bayer},
	DOI = {10.1007/978-1-4613-8431-1},
	URL = {https://doi.org/10.1007/978-1-4613-8431-1},
}






@article {ubt_polytopes,
	AUTHOR = {McMullen, P.},
	TITLE = {The maximum numbers of faces of a convex polytope},
	JOURNAL = {Mathematika},
	FJOURNAL = {Mathematika. A Journal of Pure and Applied Mathematics},
	VOLUME = {17},
	YEAR = {1970},
	PAGES = {179--184},
	ISSN = {0025-5793},
	MRCLASS = {52.10},
	MRNUMBER = {283691},
	MRREVIEWER = {G. T. Sallee},
	DOI = {10.1112/S0025579300002850},
	URL = {https://doi.org/10.1112/S0025579300002850},
}


@article {ubt_spheres,
	AUTHOR = {Stanley, Richard P.},
	TITLE = {The upper bound conjecture and {C}ohen-{M}acaulay rings},
	JOURNAL = {Studies in Appl. Math.},
	FJOURNAL = {Studies in Applied Mathematics},
	VOLUME = {54},
	YEAR = {1975},
	NUMBER = {2},
	PAGES = {135--142},
	ISSN = {0022-2526},
	MRCLASS = {57C15 (13H10 52A25)},
	MRNUMBER = {458437},
	MRREVIEWER = {P. McMullen},
	DOI = {10.1002/sapm1975542135},
	URL = {https://doi.org/10.1002/sapm1975542135},
}

@article {baby_upper,
	AUTHOR = {McMullen, P.},
	TITLE = {The numbers of faces of simplicial polytopes},
	JOURNAL = {Israel J. Math.},
	FJOURNAL = {Israel Journal of Mathematics},
	VOLUME = {9},
	YEAR = {1971},
	PAGES = {559--570},
	ISSN = {0021-2172},
	MRCLASS = {52.10},
	MRNUMBER = {278183},
	MRREVIEWER = {Joseph Zaks},
	DOI = {10.1007/BF02771471},
	URL = {https://doi.org/10.1007/BF02771471},
}

@article {several,
	AUTHOR = {McMullen, P.},
	TITLE = {The number of neighbourly {$d$}-polytopes with {$d+3$}
	vertices},
	JOURNAL = {Mathematika},
	FJOURNAL = {Mathematika. A Journal of Pure and Applied Mathematics},
	VOLUME = {21},
	YEAR = {1974},
	PAGES = {26--31},
	ISSN = {0025-5793},
	MRCLASS = {52A25},
	MRNUMBER = {367812},
	MRREVIEWER = {G. T. Sallee},
	DOI = {10.1112/S002557930000574X},
	URL = {https://doi.org/10.1112/S002557930000574X},
}

@article {simple,
	AUTHOR = {McMullen, Peter},
	TITLE = {On simple polytopes},
	JOURNAL = {Invent. Math.},
	FJOURNAL = {Inventiones Mathematicae},
	VOLUME = {113},
	YEAR = {1993},
	NUMBER = {2},
	PAGES = {419--444},
	ISSN = {0020-9910},
	MRCLASS = {52B05 (52B35)},
	MRNUMBER = {1228132},
	MRREVIEWER = {Bernd Kind},
	DOI = {10.1007/BF01244313},
	URL = {https://doi.org/10.1007/BF01244313},
}


@article {nested,
	AUTHOR = {Castillo, Federico and Liu, Fu},
	TITLE = {Deformation Cones of Nested Braid Fans},
	JOURNAL = {Int. Math. Res. Not. IMRN},
	FJOURNAL = {International Mathematics Research Notices. IMRN},
	YEAR = {2020},
	MONTH = {06},
	ISSN = {1073-7928},
	DOI = {10.1093/imrn/rnaa090},
	URL = {https://doi.org/10.1093/imrn/rnaa090},
    note = {rnaa090},
}


@misc{realization_dimension,
	title={On the Dimensions of the Realization Spaces of Polytopes},
	author={Laith Rastanawi and Rainer Sinn and Günter M. Ziegler},
	year={2020},
	eprint={2007.00645},
	archivePrefix={arXiv},
	primaryClass={math.MG}
}

@incollection {chapoton,
	AUTHOR = {Chapoton, Fr\'{e}d\'{e}ric and Fomin, Sergey and Zelevinsky, Andrei},
	TITLE = {Polytopal realizations of generalized associahedra},
	NOTE = {Dedicated to Robert V. Moody},
	JOURNAL = {Canad. Math. Bull.},
	FJOURNAL = {Canadian Mathematical Bulletin. Bulletin Canadien de
	Math\'{e}matiques},
	VOLUME = {45},
	YEAR = {2002},
	NUMBER = {4},
	PAGES = {537--566},
	ISSN = {0008-4395},
	MRCLASS = {52C07 (05E15 20F55)},
	MRNUMBER = {1941227},
	MRREVIEWER = {Alexander Zvonkin},
	DOI = {10.4153/CMB-2002-054-1},
	URL = {https://doi.org/10.4153/CMB-2002-054-1},
}

@article {fusy,
	AUTHOR = {Fusy, \'{E}ric},
	TITLE = {Counting {$d$}-polytopes with {$d+3$} vertices},
	JOURNAL = {Electron. J. Combin.},
	FJOURNAL = {Electronic Journal of Combinatorics},
	VOLUME = {13},
	YEAR = {2006},
	NUMBER = {1},
	PAGES = {Research Paper 23, 25},
}


@misc{abhy,
    title={ABHY Associahedra and Newton polytopes of $F$-polynomials for finite type cluster algebras},
    author={Véronique Bazier-Matte and Guillaume Douville and Kaveh Mousavand and Hugh Thomas and Emine Yıldırım},
    year={2018},
    eprint={1808.09986},
    archivePrefix={arXiv},
    primaryClass={math.RT}
}

@book {toric,
	AUTHOR = {Cox, David A. and Little, John B. and Schenck, Henry K.},
	TITLE = {Toric varieties},
	SERIES = {Graduate Studies in Mathematics},
	VOLUME = {124},
	PUBLISHER = {American Mathematical Society, Providence, RI},
	YEAR = {2011},
	PAGES = {xxiv+841},
	ISBN = {978-0-8218-4819-7},
	MRCLASS = {14M25 (05A15 05E45 52B12)},
	MRNUMBER = {2810322},
	MRREVIEWER = {Ivan Arzhantsev},
	DOI = {10.1090/gsm/124},
	URL = {https://doi.org/10.1090/gsm/124},
}


@article {loday,
    AUTHOR = {Loday, Jean-Louis},
     TITLE = {Realization of the {S}tasheff polytope},
   JOURNAL = {Arch. Math. (Basel)},
  FJOURNAL = {Archiv der Mathematik},
    VOLUME = {83},
      YEAR = {2004},
    NUMBER = {3},
     PAGES = {267--278},
      ISSN = {0003-889X},
   MRCLASS = {52B11 (55P47 55P48)},
  MRNUMBER = {2108555},
MRREVIEWER = {Richard John Steiner},
       DOI = {10.1007/s00013-004-1026-y},
       URL = {https://doi.org/10.1007/s00013-004-1026-y},
}


@misc{pilaud_arnau,
Author = {Arnau Padrol and Yann Palu and Vincent Pilaud and Pierre-Guy Plamondon},
Title = {Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors},
Year = {2019},
Eprint = {1906.06861v2},
Eprinttype = {arXiv},
}



@article {scattering,
    AUTHOR = {Arkani-Hamed, Nima and Bai, Yuntao and He, Song and Yan,
              Gongwang},
     TITLE = {Scattering forms and the positive geometry of kinematics,
              color and the worldsheet},
   JOURNAL = {J. High Energy Phys.},
  FJOURNAL = {Journal of High Energy Physics},
      YEAR = {2018},
    NUMBER = {5},
     PAGES = {096, front matter+75},
      ISSN = {1126-6708},
   MRCLASS = {81U20 (52B12 81T13 81U10)},
  MRNUMBER = {3832668},
MRREVIEWER = {Giorgio Cattapan},
       DOI = {10.1007/jhep05(2018)096},
       URL = {https://doi.org/10.1007/jhep05(2018)096},
}
		

@article {postnikov,
    AUTHOR = {Postnikov, Alexander},
     TITLE = {Permutohedra, associahedra, and beyond},
   JOURNAL = {Int. Math. Res. Not. IMRN},
  FJOURNAL = {International Mathematics Research Notices. IMRN},
      YEAR = {2009},
    NUMBER = {6},
     PAGES = {1026--1106},
      ISSN = {1073-7928},
   MRCLASS = {05E30},
  MRNUMBER = {2487491},
       DOI = {10.1093/imrn/rnn153},
       URL = {https://doi.org/10.1093/imrn/rnn153},
}

@manual{sagemath,
	Key          = {SageMath},
	Author       = {{The Sage Developers}},
	Title        = {{S}ageMath, the {S}age {M}athematics {S}oftware {S}ystem ({V}ersion 9.0)},
	note         = {{\tt https://www.sagemath.org}},
	Year         = {2020},
}

@manual{eg-models,
	Author       = {Günter M. Ziegler},
	Title        = {Sharir's Cube},
	note         = {Electronic Geometry Model No. 2000.09.028},
}

@misc{apr,
    title={Removahedral congruences versus permutree congruences},
    author={Doriann Albertin and Vincent Pilaud and Julian Ritter},
    year={2020},
    eprint={2006.00264},
    archivePrefix={arXiv},
    primaryClass={math.CO}
}


@incollection {mnev,
    AUTHOR = {Mn\"ev, N. E.},
     TITLE = {The universality theorems on the classification problem of
              configuration varieties and convex polytopes varieties},
 BOOKTITLE = {Topology and geometry---{R}ohlin {S}eminar},
    SERIES = {Lecture Notes in Math.},
    VOLUME = {1346},
     PAGES = {527--543},
 PUBLISHER = {Springer, Berlin},
      YEAR = {1988},
   MRCLASS = {52A25 (05B30 14G30 52A37)},
  MRNUMBER = {970093},
MRREVIEWER = {G. Ewald},
       DOI = {10.1007/BFb0082792},
       URL = {https://doi-org.offcampus.lib.washington.edu/10.1007/BFb0082792},
}