Equipartitions and Mahler volumes of symmetric convex bodies. (English) Zbl 1498.52010
Summary: Following ideas of Iriyeh and Shibata we give a short proof of the three-dimensional Mahler conjecture for symmetric convex bodies. Our contributions include, in particular, simple self-contained proofs of their two key statements. The first of these is an equipartition (ham sandwich type) theorem which refines a celebrated result of Hadwiger and, as usual, can be proved using ideas from equivariant topology. The second is an inequality relating the product volume to areas of certain sections and their duals. Finally we give an alternative proof of the characterization of convex bodies that achieve the equality case and establish a new stability result.
MSC:
| 52A38 | Length, area, volume and convex sets (aspects of convex geometry) |
References:
| [1] | LAMA, UNIV GUSTAVE EIFFEL, UNIV PARIS EST CRETEIL, CNRS, F-77447 MARNE-LA-VALLÉE, FRANCE E-mail: matthieu.fradelizi@univ-eiffel.fr UNIVERSITÉ PARIS-EST MARNE-LA-VALLÉE, LABORATOIRE D’INFORMATIQUE GASPARD MONGE, 5 BOULEVARD DESCARTES, 77420 CHAMPS-SUR-MARNE, FRANCE E-mail: alfredo.hubard@u-pem.fr |
| [2] | LAMA, UNIV GUSTAVE EIFFEL, UNIV PARIS EST CRETEIL, CNRS, F-77447 MARNE-LA-VALLÉE, FRANCE E-mail: mathieu.meyer@univ-eiffel.fr |
| [3] | CENTRO DE CIENCIAS MATEMÁTICAS, UNAM-CAMPUS MORELIA, ANTIGUA CAR- |
| [4] | RETERA A PÁTZCUARO #8701, COL. EX HACIENDA SAN JOSÉ DE LA HUERTA, MORELIA, MICHOACÁN MÉXICO, C.P. 58089 |
| [5] | E-mail: e.roldan@im.unam.mx DEPARTMENT OF MATHEMATICAL SCIENCES, KENT STATE UNIVERSITY, KENT, OH 44242 E-mail: azvavitc@kent.edu REFERENCES |
| [6] | A. Akopyan, S. Avvakumov, and R. Karasev, Convex fair partitions into an arbitrary number of pieces, preprint, http://128.84.4.18/abs/1804.03057, 2021. |
| [7] | J. C.Álvarez Paiva, F. Balacheff, and K. Tzanev, Isosystolic inequalities for optical hypersurfaces, Adv. Math. 301 (2016), 934-972. · Zbl 1388.53035 |
| [8] | S. Artstein-Avidan, A. Giannopoulos, and V. D. Milman, Asymptotic Geometric Analysis. Part I, Math. Surveys Monogr., vol. 202, American Mathematical Society, Providence, RI, 2015. · Zbl 1337.52001 |
| [9] | S. Artstein-Avidan, R. Karasev, and Y. Ostrover, From symplectic measurements to the Mahler conjecture, Duke Math. J. 163 (2014), no. 11, 2003-2022. · Zbl 1330.52004 |
| [10] | F. Barthe and M. Fradelizi, The volume product of convex bodies with many hyperplane symmetries, Amer. J. Math. 135 (2013), no. 2, 311-347. · Zbl 1277.52010 |
| [11] | P. V. M. Blagojević and A. S. Dimitrijević Blagojević, Using equivariant obstruction theory in combinato-rial geometry, Topology Appl. 154 (2007), no. 14, 2635-2655. · Zbl 1158.55009 |
| [12] | P. V. M. Blagojević, F. Frick, A. Haase, and G. M. Ziegler, Topology of the Grünbaum-Hadwiger-Ramos hyperplane mass partition problem, Trans. Amer. Math. Soc. 370 (2018), no. 10, 6795-6824. · Zbl 1396.52011 |
| [13] | K. J. Böröczky and D. Hug, Stability of the reverse Blaschke-Santaló inequality for zonoids and applica-tions, Adv. in Appl. Math. 44 (2010), no. 4, 309-328. · Zbl 1191.52002 |
| [14] | J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in R n , Invent. Math. 88 (1987), no. 2, 319-340. · Zbl 0617.52006 |
| [15] | M. Fradelizi, M. Meyer, and A. Zvavitch, An application of shadow systems to Mahler’s conjecture, Dis-crete Comput. Geom. 48 (2012), no. 3, 721-734. · Zbl 1258.52006 |
| [16] | A. Giannopoulos, G. Paouris, and B.-H. Vritsiou, The isotropic position and the reverse Santaló inequality, Israel J. Math. 203 (2014), no. 1, 1-22. · Zbl 1307.52003 |
| [17] | J. E. Goodman and J. O’Rourke (eds.), Handbook of Discrete and Computational Geometry, 2nd ed., Discrete Math. Appl. (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2004. · Zbl 1056.52001 |
| [18] | Y. Gordon, M. Meyer, and S. Reisner, Zonoids with minimal volume-product-a new proof, Proc. Amer. Math. Soc. 104 (1988), no. 1, 273-276. · Zbl 0663.52003 |
| [19] | B. Grünbaum, Partitions of mass-distributions and of convex bodies by hyperplanes, Pacific J. Math. 10 (1960), 1257-1261. · Zbl 0101.14603 |
| [20] | V. Guillemin and A. Pollack, Differential Topology. Reprint of the 1974 original, AMS Chelsea Publishing, Providence, RI, 2010. · Zbl 1420.57001 |
| [21] | H. Hadwiger, Simultane Vierteilung zweier Körper, Arch. Math. (Basel) 17 (1966), 274-278. · Zbl 0137.41501 |
| [22] | H. Iriyeh and M. Shibata, Symmetric Mahler’s conjecture for the volume product in the 3-dimensional case, Duke Math. J. 169 (2020), no. 6, 1077-1134. · Zbl 1439.52007 |
| [23] | R. Karasev, Mahler’s conjecture for some hyperplane sections, Israel J. Math. 241 (2021), no. 2, 795-815. · Zbl 1473.52015 |
| [24] | R. Karasev, A. Hubard, and B. Aronov, Convex equipartitions: the spicy chicken theorem, Geom. Dedicata 170 (2014), 263-279. · Zbl 1301.52010 |
| [25] | J. Kim, Minimal volume product near Hanner polytopes, J. Funct. Anal. 266 (2014), no. 4, 2360-2402. · Zbl 1361.52003 |
| [26] | J. Kim and A. Zvavitch, Stability of the reverse Blaschke-Santaló inequality for unconditional convex bodies, Proc. Amer. Math. Soc. 143 (2015), no. 4, 1705-1717. · Zbl 1315.52007 |
| [27] | B. Klartag, Convex geometry and waist inequalities, Geom. Funct. Anal. 27 (2017), no. 1, 130-164. · Zbl 1383.52003 |
| [28] | G. Kuperberg, From the Mahler conjecture to Gauss linking integrals, Geom. Funct. Anal. 18 (2008), no. 3, 870-892. · Zbl 1169.52004 |
| [29] | K. Mahler, Ein Minimalproblem für konvexe Polygone, Mathematica B, Zutphen 7 (1938), 118-127. · JFM 64.0732.01 |
| [30] | , Einübertragungsprinzip für konvexe Körper,Časopis Pěst. Mat. Fys. 68 (1939), 93-102. · JFM 65.0175.02 |
| [31] | V. V. Makeev, Theorems on equipartition of a continuous mass distribution, J. Math. Sci. (N.Y.) 140 (2007), no. 4, 551-557. · Zbl 1151.52304 |
| [32] | P. Mani-Levitska, S. Vrećica, and R.Živaljević, Topology and combinatorics of partitions of masses by hyperplanes, Adv. Math. 207 (2006), no. 1, 266-296. · Zbl 1110.68155 |
| [33] | J. Matoušek, Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry, Universitext, Springer-Verlag, Berlin, 2003. · Zbl 1016.05001 |
| [34] | M. Meyer, Une caractérisation volumique de certains espaces normés de dimension finie, Israel J. Math. 55 (1986), no. 3, 317-326. · Zbl 0629.46023 |
| [35] | M. Meyer and A. Pajor, On the Blaschke-Santaló inequality, Arch. Math. (Basel) 55 (1990), no. 1, 82-93. · Zbl 0718.52011 |
| [36] | M. Meyer and S. Reisner, Shadow systems and volumes of polar convex bodies, Mathematika 53 (2006), no. 1, 129-148. · Zbl 1118.52015 |
| [37] | F. Nazarov, The Hörmander proof of the Bourgain-Milman theorem, Geometric Aspects of Functional Analysis, Lecture Notes in Math., vol. 2050, Springer-Verlag, Heidelberg, 2012, pp. 335-343. · Zbl 1291.52014 |
| [38] | F. Nazarov, F. Petrov, D. Ryabogin, and A. Zvavitch, A remark on the Mahler conjecture: local minimality of the unit cube, Duke Math. J. 154 (2010), no. 3, 419-430. · Zbl 1207.52005 |
| [39] | C. M. Petty, Affine isoperimetric problems, Discrete Geometry and Convexity (New York, 1982), Ann. New York Acad. Sci., vol. 440, New York Acad. Sci., New York, 1985, pp. 113-127. · Zbl 0576.52003 |
| [40] | E. A. Ramos, Equipartition of mass distributions by hyperplanes, Discrete Comput. Geom. 15 (1996), no. 2, 147-167. · Zbl 0843.68120 |
| [41] | S. Reisner, Zonoids with minimal volume-product, Math. Z. 192 (1986), no. 3, 339-346. · Zbl 0578.52005 |
| [42] | J. Saint-Raymond, Sur le volume des corps convexes symétriques, Initiation Seminar on Analysis: G. Choquet-M. Rogalski-J. Saint-Raymond, 20th Year: 1980/1981, Publ. Math. Univ. Pierre et Marie Curie, vol. 46, Univ. Paris VI, Paris, 1981, p. Exp. No. 11. |
| [43] | L. A. Santaló, An affine invariant for convex bodies of n-dimensional space, Portugal. Math. 8 (1949), 155-161. · Zbl 0038.35702 |
| [44] | R. Schneider, Convex Bodies: the Brunn-Minkowski Theory, expanded ed., Encyclopedia Math. Appl., vol. 151, Cambridge University Press, Cambridge, 2014. · Zbl 1287.52001 |
| [45] | R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17-86. · Zbl 0057.15502 |
| [46] | F. F. Yao, D. P. Dobkin, H. Edelsbrunner, and M. S. Paterson, Partitioning space for range queries, SIAM J. Comput. 18 (1989), no. 2, 371-384 · Zbl 0675.68066 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
