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

TL;DR

This paper establishes the existence of optimal admissible meshes for certain classes of compact subsets in , including star-shaped Lipschitz domains with positive reach and  ^{1,1} domains, using polynomial norming sets and Bernstein inequalities.

Contribution

It extends the theory of optimal admissible meshes to broader classes of domains, providing constructive methods and sharp inequalities for polynomial approximation.

Findings

Existence of optimal admissible meshes for star-shaped Lipschitz domains with positive reach.

Constructive proof of optimal meshes for  ^{1,1} domains.

Use of a multivariate sharp Bernstein inequality based on the distance function.

Abstract

We show that any compact subset of $\R^d$ 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. Kro\'o on $\mathscr C^ 2$ star-shaped domains. Moreover, we prove constructively the existence of an optimal AM for any $K := \overline\Omega \subset \R^ d$ where $\Omega$ is a bounded $\mathscr C^{ 1,1}$ domain. This is done by a particular multivariate sharp version of the Bernstein Inequality via the distance function.