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This article is cited in 4 scientific papers (total in 4 papers)
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.
Received: 20.07.2017
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
Linking options:
https://www.mathnet.ru/eng/mais617 https://www.mathnet.ru/eng/mais/v25/i1/p140
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