On numerical characteristics of а simplex and their estimates

Let $n\in {\mathbb N}$, and let $Q_n=[0,1]^n$ be the $n$-dimensional unit cube. 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 the ratio of homothety $\sigma$. We apply the following numerical characteristics of the simplex. Denote by $\xi(S)$ the minimal $\sigma>0$ with the property $Q_n\subset \sigma S$. By $\alpha(S)$ we denote the minimal $\sigma>0$ such that $Q_n$ is contained in a translate of a simplex $\sigma S$. By $d_i(S)$ we mean the $i$th axial diameter of $S$, i. e. the maximum length of a segment contained in $S$ and parallel to the $i$th coordinate axis. We apply the computational formulae for $\xi(S)$, $\alpha(S)$, $d_i(S)$ which have been proved by the first author. In the paper we discuss the case $S\subset Q_n$. Let $\xi_n=\min\{ \xi(S): S\subset Q_n\}. $ Earlier the first author formulated the conjecture: if $\xi(S)=\xi_n$, then $\alpha(S)=\xi(S)$. He proved this statement for $n=2$ and the case when $n+1$ is an Hadamard number, i. e. there exists an Hadamard matrix of order $n+1$. The following conjecture is a stronger proposition: for each $n$, there exist $\gamma\geq 1$, not depending on $S\subset Q_n$, such that $\xi(S)-\alpha(S)\leq \gamma (\xi(S)-\xi_n).$ By $\varkappa_n$ we denote the minimal $\gamma$ with such a property. If $n+1$ is an Hadamard number, then the precise value of $\varkappa_n$ is 1. The existence of $\varkappa_n$ for other $n$ was unclear. In this paper with the use of computer methods we obtain an equality
$$\varkappa_2 = \frac{5+2\sqrt{5}}{3}=3.1573\ldots $$
Also we prove a new estimate
$$\xi_4\leq \frac{19+5\sqrt{13}}{9}=4.1141\ldots,$$
which improves the earlier result $\xi_4\leq \frac{13}{3}=4.33\ldots$ Our conjecture is that $\xi_4$ is precisely $\frac{19+5\sqrt{13}}{9}$. Applying this value in numerical computations we achive the value
$$\varkappa_4 = \frac{4+\sqrt{13}}{5}=1.5211\ldots$$
Denote by $\theta_n$ the minimal norm of interpolation projection on the space of linear functions of $n$ variables as an operator from $C(Q_n)$ in $C(Q_n)$. It is known that, for each $n$,
$$\xi_n\leq \frac{n+1}{2}\left(\theta_n-1\right)+1,$$
and for $n=1,2,3,7$ here we have an equality. Using computer methods we obtain the result $\theta_4=\frac{7}{3}$. Hence, the minimal $n$ such that the above inequality has a strong form is equal to 4.