On $n$-dimensional simplices satisfying inclusions $S\subset [0,1]^n\subset nS$

M. V. Nevskii ; A. Yu. Ukhalov

Modelirovanie i analiz informacionnyh sistem, Tome 24 (2017) no. 5, pp. 578-595

Résumé

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 $n\leq \xi_n$. 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.

Détail

Keywords: $n$-dimensional simplex, $n$-dimensional cube, homothety, axial diameter, interpolation, projection, numerical methods.