  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.
