Monotone twist maps and Dowker-type theorems

TL;DR

This paper links classic geometric inequalities related to inscribed and circumscribed polygons in planar ovals to the convexity of Mather's beta-function in billiard systems, deriving new inequalities for various billiards.

Contribution

It unifies classic Dowker-type theorems with Mather's beta-function convexity and introduces new geometric inequalities for billiard systems, including novel cases.

Findings

Classic inequalities are special cases of beta-function convexity.

New inequalities are derived for various billiard configurations.

Results include inequalities for higher rotation numbers.

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, Moln\'ar, 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 system. 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.