Optimal polynomial meshes exist on any multivariate convex domain (Q6566153)

The identification of sequences of finite sets of compact domains that allow to bound polynomials uniformly is of great interest in analysis and algebra. The task is to find sequences of ``small'' sets \(Y_n\) so that, for a compact domain \(\Omega\), we can estimate \(\|P\|_\Omega=\|P\|_{\infty,\Omega}\) from above as \N\[\NO(\|P\|_{Y_n})=O(\max_{Y_n}|P(\cdot)|)\N\]\Nfor all \(P\).\N\NThe \(P\)s are polynomials of some fixed total degree (call the degree \(n\)). So the sets \(Y_n=\{x_1,x_2,\ldots,x_N\}\), \(N=N(n,d)\), depend on the degree of the polynomials and of course on the dimension.\N\NThe importance lies in finding least bounds on the size of the sets \(Y_n\). In this excellent contribution an upper bound of \(O(n^d)\) is given for the cardinality of the sets, where \(d\) is the space dimension. The \(O(\cdot)\) in \(\|P\|_\Omega\) bounded by \(O(\|P\|_{Y_n})\) is even specified as 2. This settles a conjecture due to Andras Kroo.