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 $n \leqslant 118$.