Equality cases in Viterbo's conjecture and isoperimetric billiard inequalities

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.

15 pages, 5 figures. To appear in International Mathematics Research Notices. Former title "Equality cases in Viterbo's conjecture related to permutohedra"

References in corpus (2)

• Elementary approach to closed billiard trajectories in asymmetric normed spaces
• Shortest Closed Billiard Trajectories in the Plane and Equality Cases in Mahler's Conjecture

Cited by in corpus (4)

• The Viterbo's capacity conjectures for convex toric domains and the product of a $1$-unconditional convex body and its polar
• The Minkowski Billiard Characterization of the EHZ-capacity of Convex Lagrangian Products
• Coisotropic Hofer-Zehnder capacities of convex domains and related results
• From Lagrangian Products to Toric Domains via the Toda Lattice