Optimal Lagrange Interpolation Projectors and Legendre Polynomials

Let K be a convex body in Rⁿ, and let ₁ (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) ₁ (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 ₙ (K). Our main result, Theorem 5. 2, provides an explicit lower bound for the constant ₙ (K) for an arbitrary convex body K Rⁿ and an arbitrary n 1. We prove that ₙ (K) ₙ^-1 ({ vol (K) }/{ simp (K) }) where ₙ 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 1 the volume of the set \x= (x₁,. . . , xₙ) Rⁿ: |xⱼ| +|1- xⱼ|\ is equal to ₙ () /n!. If K is an n-dimensional ball, this approach leads us to the equivalence ₙ (K) which is complemented by the exact formula for ₙ (K). If K is an n-dimensional cube, we obtain explicit efficient formulae for upper and lower bounds of the constant ₙ (K) ; moreover, for small n, these estimates enable us to compute the exact values of this constant.