On Minimal Absorption Index for an n-Dimensional Simplex

Abstract

Let \(n \in \mathbb{N}\) and let \({{Q}_{n}}\) be the unit cube \({{[0,1]}^{n}}\). For a nondegenerate simplex \(S \subset {{\mathbb{R}}^{n}}\), by \(\sigma S\) denote the homothetic copy of \(S\) with center of homothety in the center of gravity of \(S\) and ratio of homothety \(\sigma .\) Put \(\xi (S) = min{\text{\{ }}\sigma \geqslant 1:{{Q}_{n}} \subset \sigma S{\text{\} }}{\text{.}}\) We call \(\xi (S)\) an absorption index of simplex \(S\). In the present paper we give new estimates for minimal absorption index of the simplex contained in \({{Q}_{n}}\), i.e., for the number \({{\xi }_{n}} = min{\text{\{ }}\xi (S):S \subset {{Q}_{n}}{\text{\} }}.\) In particular, this value and its analogues have applications in estimates for the norms of interpolation projectors. Previously the first author proved some general estimates of \({{\xi }_{n}}\). Always \(n \leqslant {{\xi }_{n}} < n + 1\). If there exist an Hadamard matrix of order \(n + 1\), then \({{\xi }_{n}} = n\). The best known general upper estimate have the form \({{\xi }_{n}} \leqslant \tfrac{{{{n}^{2}} - 3}}{{n - 1}}\)\((n > 2)\). There exist constant \(c > 0\) not depending on \(n\) such that, for any simplex \(S \subset {{Q}_{n}}\) of maximum volume, inequalities \(c\xi (S) \leqslant {{\xi }_{n}} \leqslant \xi (S)\) take place. It motivates the making use of maximum volume simplices in upper estimates of \({{\xi }_{n}}\). The set of vertices of such a simplex can be consructed with application of maximum \(0/1\)-determinant of order \(n\) or maximum \( - 1/1\)-determinant of order \(n + 1\). In the paper we compute absorption indices of maximum volume simplices in \({{Q}_{n}}\) constructed from known maximum \( - 1/1\)-determinants via special procedure. For some \(n\), this approach makes it possible to lower theoretical upper bounds of \({{\xi }_{n}}\). Also we give best known upper estimates of \({{\xi }_{n}}\) for \(n \leqslant 118\).