Higher order terms of Mather's $\beta$-function for symplectic and outer billiards

We compute explicitly the higher order terms of the formal Taylor expansion of Mather’s $\beta$-function for symplectic and outer billiards in a strictly-convex planar domain $C$. In particular, we specify the third terms of the asymptotic expansions of the distance (in the sense of the symmetric difference metric) between $C$ and its best approximating inscribed or circumscribed polygons with at most $n$ vertices. We use tools from affine differential geometry.