On n-Dimensional Simplices Satisfying Inclusions $S \subset {{[0,1]}^{n}} \subset nS$

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 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{\text{\{ }}\sigma \geqslant 1:{{Q}_{n}} \subset \sigma S{\text{\} }},$${{\xi }_{n}} = min{\text{\{ }}\xi (S):S \subset {{Q}_{n}}{\text{\} }}.$ By $\alpha (S)$ we denote the minimal $\sigma > 0$ such that ${{Q}_{n}}$ is contained in a translate of simplex $\sigma S$. Consider $(n + 1) \times (n + 1)$-matrix ${\mathbf{A}}$ with the rows containing coordinates of vertices of $S$; last column of ${\mathbf{A}}$ consists of 1’s. Put ${{{\mathbf{A}}}^{{ - 1}}}$$= ({{l}_{{ij}}})$. Denote by ${{\lambda }_{j}}$ 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 $\tfrac{1}{{{{d}_{i}}(S)}} = \tfrac{1}{2}\sum\nolimits_{j = 1}^{n + 1} \left| {{{l}_{{ij}}}} \right|,\alpha (S) = \sum\nolimits_{i = 1}^n \tfrac{1}{{{{d}_{i}}(S)}}.$ In the present paper we consider the case $S \subset {{Q}_{n}}$. Then all the ${{d}_{i}}(S) \leqslant 1$, therefore, $n \leqslant \alpha (S) \leqslant \xi (S).$ If for some simplex $S{\kern 1pt} {{'}} \subset {{Q}_{n}}$ holds $\xi (S{\kern 1pt} {{'}}) = n,$ then ${{\xi }_{n}} = n$, $\xi (S{\kern 1pt} {{'}}) = \alpha (S{\kern 1pt} {{'}})$, and ${{d}_{i}}(S{\kern 1pt} {{'}}) = 1$. However, such the simplices S ' exist not for all the dimensions $n$. The first value of $n$ with such a property is equal to $2$. For each 2-dimensional simplex, $\xi (S) \geqslant {{\xi }_{2}} = 1 + \tfrac{{3\sqrt 5 }}{5} = 2.34 \ldots > 2$. We have an estimate $n \leqslant {{\xi }_{n}} < n + 1$. The equality ${{\xi }_{n}} = n$ takes place if there exist an Hadamard matrix of order $n + 1$. Further investigation 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 $1 \leqslant n \leqslant 11$. In the first part of the paper we present some new results concerning simplices with such a condition. If $S \subset {{Q}_{n}} \subset nS$, then center of gravity of $S$ coincide with center of ${{Q}_{n}}$. We prove that $\sum\nolimits_{j = 1}^{n + 1} \left| {{{l}_{{ij}}}} \right| = 2\,(1 \leqslant i \leqslant n),\sum\nolimits_{i = 1}^n \left| {{{l}_{{ij}}}} \right| = \tfrac{{2n}}{{n + 1}}(1 \leqslant j \leqslant n + 1).$ 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 off from the cube ${{Q}_{n}}$ the equal-sized parts. Though it is true for $n = 2$ and $n = 3$, in general case this conjecture is not valid.