On continuous billiard and quasigeodesic flows characterizing alcoves and isosceles tetrahedra

TL;DR

This paper characterizes fundamental domains of affine reflection groups and isosceles tetrahedra through continuous billiard and quasigeodesic flows, linking geometric regularity with flow continuity.

Contribution

It introduces a new geometric characterization of affine reflection groups and isosceles tetrahedra via continuous flows, extending Alexandrov geometry methods.

Findings

Billiard dynamics are continuous for boundaries of class C^{2,1}.

Billiard trajectories converge to boundary geodesics under certain regularity.

Characterization of fundamental domains using continuous flows.

Abstract

We characterize fundamental domains of affine reflection groups as those polyhedral convex bodies which support a continuous billiard dynamics. We interpret this characterization in the broader context of Alexandrov geometry and prove an analogous characterization for isosceles tetrahedra in terms of continuous quasigeodesic flows. Moreover, we show an optimal regularity result for convex bodies: the billiard dynamics is continuous if the boundary is of class $\mathcal{C}^{2,1}$. In particular, billiard trajectories converge to geodesics on the boundary in this case. Our proof of the latter continuity statement is based on Alexandrov geometry methods that we discuss resp. establish first.