Equality cases in Viterbo's conjecture and isoperimetric billiard inequalities

TL;DR

This paper uses billiard techniques to identify equality cases in Viterbo's conjecture, establishing new specific cases and interpreting them as isoperimetric inequalities for billiard trajectories.

Contribution

It demonstrates that the product of a permutohedron and a simplex achieves equality in Viterbo's conjecture, providing new equality cases and insights into symplectic geometry.

Findings

Product of permutohedron and simplex yields equality in Viterbo's conjecture

New special cases of Viterbo's conjecture proved

Isoperimetric inequalities for billiard trajectories derived

Abstract

In this note we apply the billiard technique to deduce some results on Viterbo's conjectured inequality between volume of a convex body and its symplectic capacity. We show that the product of a permutohedron and a simplex (properly related to each other) delivers equality in Viterbo's conjecture. Using this result as well as previously known equality cases, we prove some special cases of Viterbo's conjecture and interpret them as isoperimetric-like inequalities for billiard trajectories.