On Some Problem for a Simplex and a Cube in ${\mathbb R}^n$

Let $S$ be a nondegenerate simplex in ${\mathbb R}^n$. Denote by $\alpha(S)$ the minimal $\sigma>0$ such that the unit cube $Q_n:=[0,1]^n$ is contained in a translate of $\sigma S$. In the case $\alpha(S)\ne 1$ the translate of $\alpha(S)S$ containing $Q_n$ is a homothetic copy of $S$ with the homothety center at some point $x\in{\mathbb R}^n$. We obtain the following computational formula for $x$. Denote by $x^{(j)}$ $(j=1,\ldots, n+1)$ the vertices of $S$. Let ${\mathbf A}$ be the matrix of order $n+1$ with the rows consisting of the coordinates of $x^{(j)};$ the last column of ${\mathbf A}$ consists of 1's. Suppose that ${\mathbf A}^{-1}=(l_{ij}).$ Then the coordinates of $x$ are the numbers
$$x_k= \frac{\sum_{j=1}^{n+1} \left(\sum_{i=1}^n \left|l_{ij}\right|\right)x^{(j)}_k -1} {\sum_{i=1}^n\sum_{j=1}^{n+1} |l_{ij}|- 2} \quad (k=1,\ldots,n).$$
Since $\alpha(S)\ne 1,$ the denominator from the right-hand part of this equality is not equal to zero. Also we give the estimates for norms of projections dealing with the linear interpolation of continuous functions defined on $Q_n$.