Estimation of Interpolation Projectors Using Legendre Polynomials

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.