Antezana, JorgeMarzo Sánchez, JordiOrtega Cerdà, Joaquim2021-09-082022-08-302021-08-301617-9447https://hdl.handle.net/2445/179888Let $\Omega$ 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 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 are no bases of orthogonal reproducing kernels in the space of polynomials of degree at most $k$.19 p.application/pdfeng(c) Springer Verlag, 2021Anàlisi harmònicaTeoria de l'aproximacióHarmonic analysisApproximation theoryinfo:eu-repo/semantics/article7137902021-09-08info:eu-repo/semantics/openAccess