Isosystolic inequalities for optical hypersurfaces

TL;DR

This paper extends systolic geometry to Finsler metrics and optical hypersurfaces, establishing bounds on periodic characteristics related to volume and connecting to the Mahler conjecture and Minkowski's theorem.

Contribution

It introduces a new systolic inequality for optical hypersurfaces in cotangent bundles, linking geometric volume to the existence of short periodic characteristics.

Findings

Periodic characteristic action is bounded by rom volume V.

A dual Minkowski theorem relates lattice points to convex body area.

New connections between Finsler geometry, Mahler conjecture, and lattice point theorems.

Abstract

We explore a natural generalization of systolic geometry to Finsler metrics and optical hypersurfaces with special emphasis on its relation to the Mahler conjecture and the geometry of numbers. In particular, we show that if an optical hypersurface of contact type in the cotangent bundle of the 2-dimensional torus encloses a volume $V$, then it carries a periodic characteristic whose action is at most $\sqrt{V/3}$. This result is deduced from an interesting dual version of Minkowski's lattice-point theorem: if the origin is the unique integer point in the interior of a planar convex body, the area of its dual body is at least 3/2.