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 $$n \leqslant 118$$ .