Effective rigidity away from the boundary for centrally-symmetric billiards

TL;DR

This paper investigates the structure of invariant sets in centrally symmetric billiard tables, providing bounds related to the shape of the boundary and characterizing the circle as the unique case of equality.

Contribution

It introduces a new invariant set within billiard tables and establishes an effective measure bound linked to the boundary's isoperimetric defect, with a characterization of the circle.

Findings

Bound on the measure of the invariant set in terms of isoperimetric defect

Equality case characterizes the circle

Provides insights into rigidity properties of billiard tables

Abstract

In this paper we study centrally symmetric Birkhoff billiard tables. We introduce a closed invariant set $\mathcal{M}_\mathcal{B}$ consisting of locally maximizing orbits of the billiard map lying inside the region $\mathcal{B}$ bounded by two invariant curves of $4$-periodic orbits. We give an effective bound from above on the measure of this invariant set in terms of the isoperimetric defect of the curve. The equality case occurs if and only if the curve is a circle.