Monotone twist maps and Dowker-type theorems

Peter Albers and Serge Tabachnikov

Vol. 330 (2024), No. 1, 1–24

DOI: 10.2140/pjm.2024.330.1

Abstract

Given a planar oval, consider the maximal area of inscribed $n$-gons resp. the minimal area of circumscribed $n$-gons. One obtains two sequences indexed by $n$, and one of Dowker’s theorems states that the first sequence is concave and the second is convex. In total, there are four such classic results, concerning areas resp. perimeters of inscribed resp. circumscribed polygons, due to Dowker, Molnár, and Eggleston. We show that these four results are all incarnations of the convexity property of Mather’s $\beta$-function (the minimal average action function) of the respective billiard-type systems. We then derive new geometric inequalities of similar type for various other billiard systems. Some of these billiards have been thoroughly studied, and some are novel. Moreover, we derive new inequalities (even for conventional billiards) for higher rotation numbers.

Keywords

monotone twist maps, Mather beta function, geometric approximation

Mathematical Subject Classification

Primary: 37C83, 52-XX

Milestones

Received: 25 February 2024

Revised: 11 June 2024

Accepted: 6 July 2024

Published: 22 July 2024