  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.
