Equipartitions and Mahler volumes of symmetric convex bodies
| Autor: | Matthieu Fradelizi, Alfredo Hubard, Mathieu Meyer, Edgardo Roldán-Pensado, Artem Zvavitch |
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| Přispěvatelé: | Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Laboratoire d'Informatique Gaspard-Monge (LIGM), École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Institut Charles Gerhardt Montpellier - Institut de Chimie Moléculaire et des Matériaux de Montpellier (ICGM), Ecole Nationale Supérieure de Chimie de Montpellier (ENSCM)-Institut de Chimie du CNRS (INC)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Instituto de Matematicas (UNAM), Universidad Nacional Autónoma de México = National Autonomous University of Mexico (UNAM), Department of Mathematical Science [Kent] (KSU-MS), Kent State University, Institut Charles Gerhardt Montpellier - Institut de Chimie Moléculaire et des Matériaux de Montpellier (ICGM ICMMM), Ecole Nationale Supérieure de Chimie de Montpellier (ENSCM)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Université Montpellier 1 (UM1)-Université Montpellier 2 - Sciences et Techniques (UM2)-Institut de Chimie du CNRS (INC), Universidad Nacional Autónoma de México (UNAM) |
| Rok vydání: | 2019 |
| Předmět: | |
| DOI: | 10.48550/arxiv.1904.10765 |
| Popis: | Following ideas of Iriyeh and Shibata we give a short proof of the three-dimensional Mahler conjecture {\mf 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. We observe that these ideas give a large family of convex sets in every dimension for which the Mahler conjecture holds true. Finally we give an alternative proof of the characterization of convex bodies that achieve the equality case and establish a {\mf new} stability result. Comment: 12 pages, 1 figure |
| Databáze: | OpenAIRE |
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