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

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.