This article is cited in
5 papers
On $n$-dimensional simplices satisfying inclusions $S\subset [0,1]^n\subset nS$
M. V. Nevskii,
A. Yu. Ukhalov P.G. Demidov Yaroslavl State University,
14 Sovetskaya str., Yaroslavl 150003, Russia
Abstract:
Let
$n\in{\mathbb N}$,
$Q_n=[0,1]^n.$
For a nondegenerate simplex
$S\subset {\mathbb R}^n$, by
$\sigma S$ we denote the homothetic image of
$S$
with the center of homothety in the center of gravity of
$S$ and ratio of homothety
$\sigma$. By
$d_i(S)$ we mean the
$i$-th axial diameter of
$S$, i. e. the maximum length of a line segment
in
$S$ parallel to the
$i$th coordinate axis.
Let
$\xi(S)=\min \{\sigma\geq 1: Q_n\subset \sigma S\},$
$\xi_n=\min \{ \xi(S): \,
S\subset Q_n \}.$
By
$\alpha(S)$ we denote the minimal
$\sigma>0$ such that
$Q_n$ is contained in a translate of simplex
$\sigma S$.
Consider
$(n+1)\times(n+1)$-matrix
$\mathbf{A}$
with the rows containing coordinates of vertices of
$S$;
the last column of
$\mathbf{A}$
consists of 1's.
Put
$\mathbf{A}^{-1}$
$=(l_{ij})$. Denote by
$\lambda_j$ a linear function on
${\mathbb R}^n$ with coefficients from the
$j$-th column of
$\mathbf{A}^{-1}$, i. e.
$\lambda_j(x)=
l_{1j}x_1+\ldots+
l_{nj}x_n+l_{n+1,j}.$
Earlier, the first author proved the equalities
$
\frac{1}{d_i(S)}=\frac{1}{2}\sum_{j=1}^{n+1} \left|l_{ij}\right|,
\ \alpha(S)
=\sum_{i=1}^n\frac{1}{d_i(S)}.$
In the present paper, we consider the case
$S\subset Q_n$.
Then all the
$d_i(S)\leq 1$, therefore,
$n\leq \alpha(S)\leq \xi(S).$
If for some simplex
$S^\prime\subset Q_n$
holds
$\xi(S^\prime)=n,$ then
$\xi_n=n$,
$\xi(S^\prime)=\alpha(S^\prime)$, and
$d_i(S^\prime)=1$.
However, such simplices
$S^\prime$ do not exist for all the dimensions
$n$. The first value of
$n$ with such a property is equal to
$2$.
For each 2-dimensional simplex,
$\xi(S)\geq \xi_2=1+\frac{3\sqrt{5}}{5}=2.34 \ldots>2$.
We have an estimate
$n\leq \xi_n<n+1$. The equality
$\xi_n=n$ takes place if
there exists an Hadamard matrix of order
$n+1$.
Further study showed that
$\xi_n=n$
also for some other
$n$.
In particular, simplices with the condition
$S\subset Q_n\subset nS$ were built for any odd
$n$ in
the interval
$1\leq n\leq 11$.
In the first part of the paper,
we present some new results concerning simplices with such a condition.
If
$S\subset Q_n\subset nS$, the center of gravity of
$S$ coincide, with the
center of
$Q_n$.
We prove that
$\sum_{j=1}^{n+1} |l_{ij}|=2 \quad (1\leq i\leq n), \ \sum_{i=1}^{n} |l_{ij}|=\frac{2n}{n+1} \ (1\leq j\leq n+1).$
Also we give some corollaries.
In the second part of the paper, we consider the following conjecture.
Let for simplex $S\subset Q_n$ an equality $\xi(S)=\xi_n$ holds.
Then $(n-1)$-dimensional
hyperplanes containing the faces of $S$ cut from the cube
$Q_n$ the equal-sized parts.
Though it is true for
$n=2$ and
$n=3$, in the general case this conjecture is not valid.
Keywords:
$n$-dimensional simplex, $n$-dimensional cube, homothety, axial diameter,
interpolation, projection, numerical methods.
UDC:
514.17+
517.51+
519.6 Received: 10.02.2017
DOI:
10.18255/1818-1015-2017-5-578-595