Optimal Lagrange Interpolation Projectors and Legendre Polynomials

Let $K$ be a convex body in ${\mathbb R}^n$, and let $\Pi_1({\mathbb R}^n)$ be the space of polynomials in $n$ variables of degree at most $1$. Given an $(n+1)$-element set $Y\subset K$ in general position, we let $P_Y$ denote the Lagrange interpolation projector $P_Y: C(K)\to \Pi_1({\mathbb R}^n)$ with nodes in $Y$. In this paper, we study upper and lower bounds for the norm of the optimal Lagrange interpolation projector, i.e., the projector with minimal operator norm where the minimum is taken over all $(n+1)$-element sets of interpolation nodes in $K$. We denote this minimal norm by $\theta_n(K)$. Our main result, Theorem 5.2, provides an explicit lower bound for the constant $\theta_n(K)$ for an arbitrary convex body $K\subset{\mathbb R}^n$ and an arbitrary $n\ge 1$. We prove that $\theta_n(K)\ge \chi_n^{-1}\left({{\rm vol}(K)}/{{\rm simp}(K)}\right)$ where $\chi_n$ is the Legendre polynomial of degree $n$ and ${\rm simp}(K)$ is the maximum volume of a simplex contained in $K$. The proof of this result relies on a geometric characterization of the Legendre polynomials in terms of the volumes of certain convex polyhedra. More specifically, we show that for every $\gamma\ge 1$ the volume of the set $\left\{x=(x_1,...,x_n)\in{\mathbb R}^n : \sum |x_j| +\left|1- \sum x_j\right|\le\gamma\right\}$ is equal to ${\chi_n(\gamma)}/{n!}$. If $K$ is an $n$-dimensional ball, this approach leads us to the equivalence $\theta_n(K) \asymp\sqrt{n}$ which is complemented by the exact formula for $\theta_n(K)$. If $K$ is an $n$-dimensional cube, we obtain explicit efficient formulae for upper and lower bounds of the constant $\theta_n(K)$; moreover, for small $n$, these estimates enable us to compute the exact values of this constant.