On n-Dimensional Simplices Satisfying Inclusions S ⊂ [0, 1]<sup>n</sup> ⊂ 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 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 \({\bf A}\) with the rows containing coordinates of vertices of \(S\); the last column of \({\bf A}\) consists of 1's. Put \({\bf A}^{-1}\) \(=(l_{ij})\). Denote by \(\lambda_j\) a linear function on \({\mathbb R}^n\) with coefficients from the \(j\)-th column of \({\bf A}^{-1}\), 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.