Equipartitions and Mahler volumes of symmetric convex bodies

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 w...

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Bibliographic Details
Published inAmerican journal of mathematics Vol. 144; no. 5; pp. 1201 - 1219
Main Authors Fradelizi, Matthieu, Hubard, Alfredo, Meyer, Mathieu, Roldán-Pensado, Edgardo, Zvavitch, Artem
Format Journal Article
LanguageEnglish
Published Baltimore Johns Hopkins University Press 01.10.2022
Subjects
Functional Analysis
Mathematics
Topology
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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.
ISSN:0002-9327
1080-6377
1080-6377
DOI:10.1353/ajm.2022.0027