Equipartitions and Mahler Volumes of Symmetric Convex Bodies

Matthieu Fradelizi (1) , Alfredo Hubard (2) , Mathieu Meyer (3) , Edgardo Roldán-Pensado (4) , Artem Zvavitch (5)

1 LAMA - Laboratoire Analyse et de Mathématiques Appliquées
2 LIGM - Laboratoire d'Informatique Gaspard-Monge
3 ICGM - Institut Charles Gerhardt Montpellier - Institut de Chimie Moléculaire et des Matériaux de Montpellier
4 UNAM - Instituto de Matematicas
5 KSU-MS - Department of Mathematical Science [Kent]

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.

Domaines: Analyse fonctionnelle [math.FA]

Matthieu Fradelizi, Alfredo Hubard, Mathieu Meyer, Edgardo Roldán-Pensado, Artem Zvavitch. Equipartitions and Mahler Volumes of Symmetric Convex Bodies. American Journal of Mathematics, 2022, 144 (5), pp.1201-1219. ⟨10.1353/ajm.2022.0027⟩. ⟨hal-02924423⟩