New Estimates of Numerical Values Related to a Simplex - Mikhail V. Nevskii; Alexey Yu. Ukhalov (Моделирование и анализ информационных систем, 2017)
Let \(n\in {\mathbb N}\) and \(Q_n=[0,1]^n\). For a nondegenerate simplex \(S\subset {\mathbb R}^n\), by \(σ S\) we denote the homothetic copy of~\(S\) with center of homothety in the center of gravity of \(S\) and ratio of~homothety \(σ\). By \(ξ(S)\) we mean the minimal \(σ>0\) such that \(Q_n\subset σ S\). By \(α(S)\) denote the minimal \(σ>0\) such that \(Q_n\) is~contained in a translate of~\(σ S\). 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~\(ξ(S)\), \(α(S)\), \(d_i(S)\) were proved earlier by the first author. Define \(ξ_n=\min\{ ξ(S): S\subset Q_n\}. \) We always have \(ξ_n≥ n.\) We discuss some conjectures formulated in the previous papers. One of~these conjectures is the following. For~every \(n\), there exists \(γ>0\), not depending on \(S\subset Q_n\), such that an~inequality \(ξ(S)-α(S)≤ γ (ξ(S)-ξ_n)\) holds. Denote by \(\varkappa_n\) the minimal \(γ\) with such a~property. We prove that \(\varkappa_1=\frac{1}{2}\); for \(n>1\), we obtain \(\varkappa_n≥ 1\). If \(n>1\) and \(ξ_n=n,\) then \(\varkappa_n=1\). The equality \(ξ_n=n\) 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 \(ξ_5=5\). Therefore, there exists \(n\) such that \(n+1\) is not an Hadamard number and nevertheless \(ξ_n=n\). 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 \(ξ_n=n.\) 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 \(α(S)=ξ(S)=5.\) These simplices were found with the use of numerical and symbolic computations. %Numerical experiments allow to discover Another new result is an inequality \(ξ_6\