  Let $S$ be a set of $n$ points in general position in the plane, and let
$X_{k,\ell}(S)$ be the number of convex $k$-gons with vertices in $S$ that have
exactly $\ell$ points of $S$ in their interior. We prove several equalities for
the numbers $X_{k,\ell}(S)$. This problem is related to the Erd\H{o}s-Szekeres
theorem. Some of the obtained equations also extend known equations for the
numbers of empty convex polygons to polygons with interior points. Analogous
results for higher dimension are shown as well.

Combinatorics