Estimation of interpolation projectors using Legendre polynomials

We give some estimates for the minimum projector norm under linear interpolation on a compact set in Rⁿ. Let Π₁( Rⁿ) be the space of polynomials in n variables of degree at most $1$, Ω is a compactum in Rⁿ, K= conv(Ω). We will assume that vol(K)>0. Let the points x⁽ʲ⁾∈ Ω, 1≤ j≤ n+1, be the vertices of an n-dimensional nondegenerate simplex. The interpolation projector P:C(Ω)→ Π₁( Rⁿ) with the nodes x⁽ʲ⁾ is defined by the equations Pf(x⁽ʲ⁾)=f(x⁽ʲ⁾). By \|P\|_Ω we mean the norm of P as an operator from C(Ω) to C(Ω). By θₙ(Ω) we denote the minimal norm \|P\|_Ω of all operators P with nodes belonging to Ω. By simp(E) we denote the maximal volume of the simplex with vertices in E. We establish the inequalities χₙ⁻¹( vol(K) simp(Ω))≤ θₙ(Ω)≤ n+1. Here χₙ 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 γ≥ 1 the volume of the set =(x₁,...,xₙ)∈ Rⁿ : ∑ |xⱼ| +|1- ∑ xⱼ|≤γ\ is equal to χₙ(γ)/n!. In the case when Ω is an n-dimensional cube or an n-dimensional ball, the lower estimate gives the possibility to obtain the inequalities of the form θₙ(Ω) c√n. Also we formulate some open questions.