  Let $\Omega$ be a convex open set in $\mathbb R^n$ and let $\Lambda_k$ be a
finite subset of $\Omega$. We find necessary geometric conditions for
$\Lambda_k$ to be interpolating for the space of multivariate polynomials of
degree at most $k$. Our results are asymptotic in $k$. The density conditions
obtained match precisely the necessary geometric conditions that sampling sets
are known to satisfy, and they are expressed in terms of the equilibrium
potential of the convex set. Moreover, we prove that in the particular case of
the unit ball, for $k$ large enough, there is no family of orthogonal
reproducing kernels in the space of polynomials of degree at most $k$.
