Duality of Caustics in Minkowski Billiards

TL;DR

This paper investigates the duality of convex caustics in Minkowski billiards, revealing a strong duality in Euclidean cases and demonstrating its failure in general Minkowski settings.

Contribution

It establishes a duality relationship for convex caustics in Euclidean Minkowski billiards and shows that this duality does not extend to all Minkowski billiards.

Findings

Dual convex caustics exist in Euclidean Minkowski billiards with matching properties.

The duality phenomenon fails in general Minkowski billiards, allowing caustics without duals.

Euclidean billiards exhibit a strong duality in caustic properties.

Abstract

In this paper we study convex caustics in Minkowski billiards. We show that for the Euclidean billiard dynamics in a planar smooth centrally symmetric and strictly convex body $K$, for every convex caustic which $K$ possesses, the "dual" billiard dynamics in which the table is the Euclidean unit disk and the geometry that governs the motion is induced by the body $K$, possesses a dual convex caustic. Such a pair of caustics is dual in a strong sense, and in particular they have the same perimeter, Lazutkin parameter (both measured with respect to the corresponding geometries), and rotation number. We show moreover that for general Minkowski billiards this phenomenon fails, and one can construct a smooth caustic in a Minkowski billiard table which possesses no dual convex caustic.