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 \(\xi(S)=\min \{\sigma\geq 1: Q_n\subset \sigma S\}.\) We call \(\xi(S)\) an absorption index of simplex \(S\). In the present paper, we give new estimates for the minimal absorption index of the simplex contained in \(Q_n\), i.\,e., for the number \(\xi_n=\min \{ \xi(S): , S\subset Q_n \}.\) 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 \(n\leq\xi_n 2)\). There exists a constant \(c>0\) not depending on \(n\) such that, for any simplex \(S\subset Q_n\) of maximum volume, inequalities \(c\xi(S)\leq \xi_n\leq \xi(S)\) take place. It motivates the 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 a 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 \(n\leq 118\)