2022/04/13 by Daniel Tsodikovich, Tsodikovich, Daniel
Biochemistry, Genetics and Molecular Biology · Mathematics · #37C83 #52A40 #Diffusion and Search Dynamics #Dynamical Systems (math.DS) #FOS: Mathematics #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2204.06209
openalex publication_date 2022/04/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Blaschke-Santaló 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-Santaló 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.