Billiards in convex bodies with acute angles

TL;DR

This paper proves the existence of closed billiard trajectories in convex bodies with certain acute-angle tangent cones at non-smooth boundary points, extending billiard dynamics to less regular convex shapes.

Contribution

It establishes the existence of closed billiard trajectories in convex bodies with non-smooth boundaries under an acute tangent cone condition, a novel generalization.

Findings

Closed billiard trajectories exist in convex bodies with acute tangent cones.

The result applies to non-smooth convex bodies, broadening previous smoothness assumptions.

Provides new insights into billiard dynamics in irregular convex shapes.

Abstract

In this paper we investigate the existence of closed billiard trajectories in not necessarily smooth convex bodies. In particular, we show that if a body $K\subset \mathbb{R}^d$ has the property that the tangent cone of every non-smooth point $q\in \partial K$ is acute (in a certain sense) then there is a closed billiard trajectory in $K$.