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 Model. Anal. Inform. Sist., 2018, Volume 25, Number 1, Pages 140–150 (Mi mais617)

Computational Geometry

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

M. V. Nevskii, A. Yu. Ukhalov

Centre of Integrable Systems, P.G. Demidov Yaroslavl State University, 14 Sovetskaya str., Yaroslavl, 150003, Russian Federation

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 $n\leq\xi_n< n+1$. 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 $n\leq 118$.

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

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 1.10160.2017/5.1 This work was carried out within the framework of the state programme of the Ministry of Education and Science of the Russian Federation, project ¹ 1.10160.2017/5.1.

DOI: https://doi.org/10.18255/1818-1015-2018-1-140-150

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UDC: 514.17+517.51+519.6

Citation: M. V. Nevskii, A. Yu. Ukhalov, “On minimal absorption index for an $n$-dimensional simplex”, Model. Anal. Inform. Sist., 25:1 (2018), 140–150

Citation in format AMSBIB
\Bibitem{NevUkh18} \by M.~V.~Nevskii, A.~Yu.~Ukhalov \paper On minimal absorption index for an $n$-dimensional simplex \jour Model. Anal. Inform. Sist. \yr 2018 \vol 25 \issue 1 \pages 140--150 \mathnet{http://mi.mathnet.ru/mais617} \crossref{https://doi.org/10.18255/1818-1015-2018-1-140-150} \elib{http://elibrary.ru/item.asp?id=32482547} 

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This publication is cited in the following articles:
1. M. V. Nevskii, A. Yu. Ukhalov, “On optimal interpolation by linear functions on an $n$-dimensional cube”, Automatic Control and Computer Sciences, 52:7 (2018), 828–842
2. M. V. Nevskii, “O nekotorykh zadachakh dlya simpleksa i shara v ${\mathbb R}^n$”, Model. i analiz inform. sistem, 25:6 (2018), 680–691
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