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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.
UDC:
514.17+
517.51+
519.6 Received: 20.07.2017
DOI:
10.18255/1818-1015-2018-1-140-150
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