# A Brunn-Minkowski type inequality for extended symplectic capacities of convex domains and length estimate for a class of billiard trajectories

## TL;DR

This paper extends a key inequality in symplectic geometry to broader capacities of convex domains and establishes new length estimates for a class of non-periodic billiard trajectories within these domains.

## Contribution

It generalizes the Brunn-Minkowski inequality for symplectic capacities and introduces length estimates for non-periodic billiard trajectories in convex domains.

## Key findings

- Extended the Brunn-Minkowski inequality to new symplectic capacities.
- Proved length estimates for non-periodic billiard trajectories.
- Established parallels between periodic and non-periodic billiard results.

## Abstract

In this paper, we firstly generalize the Brunn-Minkowski type inequality for Ekeland-Hofer-Zehnder symplectic capacity of bounded convex domains established by Artstein-Avidan-Ostrover in 2008 to extended symplectic capacities of bounded convex domains constructed by authors based on a class of Hamiltonian non-periodic boundary value problems recently. Then we introduce a class of non-periodic billiards in convex domains, and for them we prove some corresponding results to those for periodic billiards in convex domains obtained by Artstein-Avidan-Ostrover in 2012.