Let \(\Omega \) be a smooth, bounded, convex domain 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 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 are no bases of orthogonal reproducing kernels in the space of polynomials of degree at most k.

令\(\Omega \)是\({\mathbb {R}}^n\) 中的一个平滑的、有界的凸域，并让\(\Lambda _k\)是\(\Omega \)的有限子集。我们发现\(\Lambda _k\) 的必要几何条件是对至多k的多元多项式空间进行插值。我们的结果在k 中是渐近的。获得的密度条件与已知采样集满足的必要几何条件精确匹配，并以凸集的平衡势表示。此外，我们证明，在单位球的特殊情况下，对于k足够大，在最多为k的多项式空间中没有正交再现核的基。