Some properties of $0/1$-simplices

Let $n\in {\mathbb N}$, and let $Q_n=[0,1]^n$. For a nondegenerate simplex $S\subset {\mathbb R}^n$, by $\sigma S$ we mean the homothetic copy of $S$ with center of homothety in the center of gravity of $S$ and ratio of homothety $\sigma$. Put $\xi(S)=\min\{\sigma\geq 1: Q_n\subset \sigma S\}$, $\xi_n=\min\{\xi(S): S\subset Q_n\}$. By $P$ we denote the interpolation projector from $C(Q_n)$ onto the space of linear functions of $n$ variables with the nodes in the vertices of a simplex $S\subset Q_n$. Let $\|P\|$ be the norm of $P$ as an operator from $C(Q_n)$ to $C(Q_n)$, $\theta_n=\min\|P\|$. By $\xi^\prime_n$ and $\theta^\prime_n$ we denote the values analogous to $\xi_n$ and $\theta_n$, with the additional condition that corresponding simplices are $0/1$-polytopes, i. e., their vertices coincide with vertices of $Q_n$. In the present paper, we systematize general estimates of the numbers $\xi^\prime_n$, $\theta^\prime_n$ and also give their new estimates and precise values for some $n$. We prove that $\xi^\prime_n\asymp n$, $\theta^\prime_n\asymp \sqrt{n}$. Let one vertex of $0/1$-simplex $S^*$ be an arbitrary vertex $v$ of $Q_n$ and the other $n$ vertices are close to the vertex of the cube opposite to $v$. For $2\leq n\leq 5$, each simplex extremal in the sense of $\xi^\prime_n$ coincides with $S^*$. The minimal $n$ such that $\xi(S^*)>\xi^\prime_n$ is equal to $6$. Denote by $P^*$ the interpolation projector with the nodes in the vertices of $S^*$. The minimal $n$ such that $\|P^*\|>\theta^\prime_n$ is equal to $5$.