Optimal polynomial admissible meshes on some classes of compact subsets of RdRd

We show that any compact subset of RdRd which is the closure of a bounded star-shaped Lipschitz domain ΩΩ, such that ∁Ω∁Ω has positive reach in the sense of Federer, admits an optimal AM   (admissible mesh), that is a sequence of polynomial norming sets with optimal cardinality. This extends a recent result of A. Kroó on C2C2 star-shaped domains.Moreover, we prove constructively the existence of an optimal AM for any K:=Ω¯⊂Rd where ΩΩ is a bounded C1,1C1,1 domain. This is done by a particular multivariate sharp version of the Bernstein Inequality via the distance function.