On minimal absorption index for an $n$-dimensional simplex

M. V. Nevskii ; A. Yu. Ukhalov

Modelirovanie i analiz informacionnyh sistem, Tome 25 (2018) no. 1, pp. 140-150

Résumé

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 $n\leq\xi_n n+1$. If there exists an Hadamard matrix of order $n+1$, then $\xi_n=n$. The best known general upper estimate has the form $\xi_n\leq \frac{n^2-3}{n-1}$ $(n>2)$. 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 $n\leq 118$.

Détail

Keywords: $n$-dimensional simplex, $n$-dimensional cube, homothety, numerical methods.
Mots-clés : absorption index, interpolation