On Numerical Characteristics of а Simplex and their Estimates

Let \(n\in {\mathbb N}\), and let \(Q_n=[0,1]^n\) be the \(n\)-dimensionalunit 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 theratio of homothety \(\sigma\). We apply the followingnumerical 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 translateof 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 parallelto the \(i\)th coordinate axis. We apply the computationalformulae for\(\xi(S)\), \(\alpha(S)\), \(d_i(S)\) which have been proved by the firstauthor. 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:{\it if\(\xi(S)=\xi_n\), then \(\alpha(S)=\xi(S)\).} He proved this statementfor \(n=2\) and the case when \(n+1\) is an Hadamard number, i.\,e.there exists an Hadamard matrix of order \(n+1\). The followingconjecture is a strongerproposition: {\it 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 obtainan 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 valuein numerical computations we achive the value$$\varkappa_4 = \frac{4+\sqrt{13}}{5}=1.5211\ldots$$Denote by \(\theta_n\) the minimal normof 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 formis equal to 4.