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

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.