On n-Dimensional Simplices Satisfying Inclusions S ⊂ [0, 1]ⁿ ⊂ 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 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.

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