Symplectic polarity and Mahler's conjecture

TL;DR

This paper explores a conjecture linking symplectic polarity with Mahler's conjecture, establishing equivalences and bounds for symplectically self-polar convex bodies to advance understanding in convex and symplectic geometry.

Contribution

It introduces a new conjecture connecting symplectic polarity to Mahler's conjecture and provides bounds for symplectic capacities of symplectically self-polar bodies.

Findings

Conjecture about volume of symplectically self-polar convex bodies

Equivalence between this conjecture and Mahler's conjecture

Bounds for symplectic capacities of these bodies

Abstract

We state a conjecture about the volume of symplectically self-polar convex bodies and show that it is equivalent to Mahler's conjecture concerning the volume of a convex body and its Euclidean polar. We also establish lower and upper bounds for symplectic capacities of symplectically self-polar bodies.