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

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.