  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.
