Estimation of interpolation projectors using Legendre polynomials

We give some estimates for the minimum projector norm under linear interpolation on a compact set in ${\mathbb R}^n$. Let $\Pi_1({\mathbb R}^n)$ be the space of polynomials in $n$ variables of degree at most $1$, $\Omega$ is a compactum in ${\mathbb R}^n$, $K={\mathrm conv}(\Omega)$. We will assume that ${\mathrm vol}(K)\geq 0$. Let the points $x^{(j)}\in \Omega$, $1\leq j\leq n+1,$ 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 equations $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$. By $\|P\|_\Omega$ we mean the norm of $P$ as an operator from $C(\Omega)$ to $C(\Omega)$. By $\theta_n(\Omega)$ we denote the minimal norm $\|P\|_\Omega$ of all operators $P$ with nodes belonging to $\Omega$. By ${\mathrm simp}(E)$ we denote the maximal volume of the simplex with vertices in $E$. We establish the inequalities $\chi_n^{-1}\left(\frac{{\mathrm vol}(K)}{{\mathrm simp}(\Omega)}\right)\leq \theta_n(\Omega)\leq n+1.$ Here $\chi_n$ is the standardized Legendre polynomial of degree $n$. The lower estimate is proved using the obtained characterization of Legendre polynomials through the volumes of convex polyhedra. More specifically, we show that for every $\gamma\ge 1$ the volume of the set $\left\{x=(x_1,\dots ,x_n)\in{\mathbb R}^n : \sum |x_j| +\left|1- \sum x_j\right|\le\gamma\right\}$ is equal to ${\chi_n(\gamma)}/{n!}$. 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.