We show that any compact subset of $\R^d$ which is the closure of a boundedstar-shaped Lipschitz domain $\Omega$, such that $\complement \Omega$ haspositive reach in the sense of Federer, admits an \emph{optimal AM} (admissiblemesh), 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-shapeddomains. 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 theBernstein Inequality via the distance function.
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机译：我们证明，$ \ R ^ d $的任何紧凑子集（是有界星状Lipschitz域$ \ Omega $的闭合），使得$ \ complement \ Omega $在费德勒的意义上具有正的影响，承认\ emph {最佳AM}（可允许网格），它是具有最佳基数的多项式范数集的序列。这扩展了A. Kro \ o在$ mathscr C ^ 2 $星型域上的最新结果。此外，我们以建设性的方式证明了任何$ K：= \ overline \ Omega \ subset \ R ^ d $的最优AM的存在，其中$ \ Omega $是有界的$ \ mathscr C ^ {1,1} $域。这是通过距离函数通过伯恩斯坦不等式的特定多元尖锐形式完成的。
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