Optimal Lagrange Interpolation Projectors and Legendre Polynomials

Let K be a convex body in Rⁿ, and let Π₁( Rⁿ) be the space of polynomials in n variables of degree at most $1$. Given an $(n+1)$-element set Y⊂ K in general position, we let PY denote the Lagrange interpolation projector PY: C(K)→ Π₁( Rⁿ) 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 θₙ(K). Our main result, Theorem 5.2, provides an explicit lower bound for the constant θₙ(K) for an arbitrary convex body K⊂ Rⁿ and an arbitrary n≥ 1. We prove that θₙ(K)≥ χₙ⁻¹( vol(K)/ simp(K)) where χₙ is the Legendre polynomial of degree n and 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 γ≥ 1 the volume of the set =(x₁,...,xₙ)∈ Rⁿ : ∑ |xⱼ| +|1- ∑ xⱼ|≤γ\ is equal to χₙ(γ)/n!. If K is an n-dimensional ball, this approach leads us to the equivalence θₙ(K) √n which is complemented by the exact formula for θₙ(K). If K is an n-dimensional cube, we obtain explicit efficient formulae for upper and lower bounds of the constant θₙ(K); moreover, for small n, these estimates enable us to compute the exact values of this constant.