Given a convex billiard table, one defines the set  swept by locally maximizing orbits for convex billiard. This is a remarkable closed invariant set which does not depend (under certain assumptions) on the choice of the generating function. I shall show how to get sharp estimates on the measure of this set, recovering as a corollary rigidity result for centrally symmetric convex billiards. Also I shall discuss rigidity of Mather β function. Based on joint works with Andrey E. Mironov, Sergei Tabachnikov and Daniel Tsodikovich