Optimal Polynomial Admissible Meshes on Some Classes of Compact Subsets of ${\mathbb{R}}^d$

Abstract

We show that any compact subset of ${\mathbb{R}}^d$ which is the closure of a bounded star-shaped Lipschitz domain $\Omega$, such that $\complement \Omega$ has positive reach in the sense of Federer, admits an \emph{optimal AM} (admissible mesh), that is a sequence of polynomial norming sets with optimal cardinality. This extends a recent result of A. Kroó on ${\mathcal{C}}^2$ star-shaped domains. Moreover, we prove constructively the existence of an optimal AM for any $K:=\overline{\Omega }\subset {\mathbb{R}}^d$ where $\Omega$ is a bounded ${\mathcal{C}}^{1,1}$ domain. This is done by a particular multivariate sharp version of the Bernstein Inequality via the distance function.

29 pages, 3figures

References in corpus (1)

• Weighted Pluripotential Theory Results of Berman-Boucksom

Cited by in corpus (1)

• Banach spaces of polynomials as "large" subspaces of ${\mathbf{\ell }}^{\infty }$-spaces