An analogue of the Blaschke-Santal\'{o} inequality for billiard dynamics

TL;DR

This paper explores an analogue of the Blaschke-Santaló inequality within billiard dynamics, relating the shortest closed billiard trajectory length to convex body properties, and establishes explicit formulas and bounds in the planar case.

Contribution

It introduces the billiard product as an analogue to volume product and derives explicit formulas and bounds for it in planar convex bodies.

Findings

Explicit expression for the billiard product in terms of diameter.

Upper bounds for the billiard product in polygons with fixed vertices.

Connection between billiard dynamics and convex geometric inequalities.

Abstract

The Blaschke-Santal\'{o} inequality is a classical inequality in convex geometry concerning the volume of a convex body and that of its dual. In this work we investigate an analogue of this inequality in the context of billiard dynamical system: we replace the volume with the length of the shortest closed billiard trajectory. We define a quantity called the "billiard product" of a convex body K, which is analogous to the volume product studied in the Blaschke-Santal\'{o} inequality. In the planar case, we derive an explicit expression for the billiard product in terms of the diameter of the body. We also investigate upper bounds for this quantity in the class of polygons with a fixed number of vertices.