Abstract
We give some estimates for the minimal projector norm under linear interpolation on a compact subset of \({{\mathbb{R}}^{n}}\) . Let \({{\Pi }_{1}}({{\mathbb{R}}^{n}})\) be the space of polynomials in \(n\) variables of degree at most \(1\) , \(\Omega \) is a compactum in \({{\mathbb{R}}^{n}}\) , and \(K = {\text{conv}}(\Omega )\) . We will assume that \({\text{vol}}(K) > 0\) . Let \({{x}^{{(j)}}} \in \Omega \) , \(1 \leqslant j \leqslant n + 1,\) be the vertices of an \(n\) -dimensional nondegenerate simplex. The interpolation projector \(P:C(\Omega ) \to {{\Pi }_{1}}({{\mathbb{R}}^{n}})\) with the nodes \({{x}^{{(j)}}}\) is defined by the equalities \(Pf\left( {{{x}^{{(j)}}}} \right) = f\left( {{{x}^{{(j)}}}} \right)\) . By \({{\left\| P \right\|}_{\Omega }}\) we mean the norm of \(P\) as an operator from \(C(\Omega )\) to \(C(\Omega )\) . By \({{\theta }_{n}}(\Omega )\) we denote the minimal norm \({{\left\| P \right\|}_{\Omega }}\) of all operators \(P\) with nodes belonging to \(\Omega \) . Let \({\text{sim}}{{{\text{p}}}_{n}}(\Omega )\) be the maximum volume of a simplex with vertices in \(\Omega \) . We establish the inequalities \(\chi _{n}^{{ - 1}}\left( {\frac{{{\text{vol}}(K)}}{{{\text{sim}}{{{\text{p}}}_{n}}(\Omega )}}} \right) \leqslant {{\theta }_{n}}(\Omega ) \leqslant n + 1.\) Here \({{\chi }_{n}}\) is the standardized Legendre polynomial of degree \(n\) . The lower estimate is proved using the obtained characterization of the Legendre polynomials through the volumes of convex polyhedra. More specifically, we show that for every \(\gamma \geqslant 1\) the volume of the set \(\left\{ {x = ({{x}_{1}},...,{{x}_{n}}) \in {{\mathbb{R}}^{n}}:\sum \left| {{{x}_{j}}} \right| + \left| {1 - \sum {{x}_{j}}} \right| \leqslant \gamma } \right\}\) is equal to \({{\chi }_{n}}(\gamma ){\text{/}}n!\) . In the case when \(\Omega \) is an \(n\) -dimensional cube or an \(n\) -dimensional ball, the lower estimate gives the possibility to obtain the inequalities of the form \({{\theta }_{n}}(\Omega ) \geqslant c\sqrt n \) . Also we formulate some open questions.
