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

We show that any compact subset of ${\mathbb{R}}^d$ which is the closure of a bounded star-shaped Lipschitz domain $\mathrm{\Omega }$, such that $\mathrm{\complement }\mathrm{\Omega }$ 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 ${\mathcal{C}}^2$ star-shaped domains. Moreover, we prove constructively the existence of an optimal AM for any $K:=\bar{\mathrm{\Omega }}\subset {\mathbb{R}}^d$ where $\mathrm{\Omega }$ 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.