New estimates of numerical values related to a simplex

Let $n\in {\mathbb N}$ and $Q_n=[0,1]^n$. For a nondegenerate simplex $S\subset {\mathbb R}^n$, by $\sigma S$ we denote the homothetic copy of $S$ with center of homothety in the center of gravity of $S$ and ratio of homothety $\sigma$. By $\xi(S)$ we mean the minimal $\sigma>0$ such that $Q_n\subset \sigma S$. By $\alpha(S)$ denote the minimal $\sigma>0$ such that $Q_n$ is contained in a translate of $\sigma S$. By $d_i(S)$ we denote the $i$th axial diameter of $S$, i. e. the maximum length of the segment contained in $S$ and parallel to the $i$th coordinate axis. Formulae for $\xi(S)$, $\alpha(S)$, $d_i(S)$ were proved earlier by the first author. Define $\xi_n=\min\{ \xi(S): S\subset Q_n\}. $ We always have $\xi_n\geq n.$ We discuss some conjectures formulated in the previous papers. One of these conjectures is the following. For every $n$, there exists $\gamma>0$, not depending on $S\subset Q_n$, such that an inequality $\xi(S)-\alpha(S)\leq \gamma (\xi(S)-\xi_n)$ holds. Denote by $\varkappa_n$ the minimal $\gamma$ with such a property. We prove that $\varkappa_1=\frac{1}{2}$; for $n>1$, we obtain $\varkappa_n\geq 1$. If $n>1$ and $\xi_n=n,$ then $\varkappa_n=1$. The equality $\xi_n=n$ holds if $n+1$ is an Hadamard number, i. e. there exists an Hadamard matrix of order $n+1$. This proposition is known; we give one more proof with the direct use of Hadamard matrices. We prove that $\xi_5=5$. Therefore, there exists $n$ such that $n+1$ is not an Hadamard number and nevertheless $\xi_n=n$. The minimal $n$ with such a property is equal to $5$. This involves $\varkappa_5=1$ and also disproves the following previous conjecture of the first author concerning the characterization of Hadamard numbers in terms of homothety of simplices: $n+1$ is an Hadamard number if and only if $\xi_n=n$. This statement is valid only in one direction. There exists a simplex $S\subset Q_5$ such that the boundary of the simplex $5S$ contains all the vertices of the cube $Q_5$. We describe a one-parameter family of simplices contained in $Q_5$ with the property $\alpha(S)=\xi(S)=5.$ These simplices were found with the use of numerical and symbolic computations. Another new result is an inequality $\xi_6\ <6.0166$. We also systematize some of our estimates of numbers $\xi_n$, $\theta_n$, $\varkappa_n$ derived by now. The symbol $\theta_n$ denotes the minimal norm of interpolation projection on the space of linear functions of $n$ variables as an operator from $C(Q_n)$ to $C(Q_n)$.