Properties of axial diameters of a simplex (Q635760)

Let \(S\) be a nondegenerate simplex in \(\mathbb R^{n }\). It is proved that the minimal possible \(\sigma >0\), such that a homothetic copy of \(S\) of ratio \(\sigma \) contains \([0,1]^{n }\), is equal to \(\sum_{i=1}^{n} 1/d_{i}(S)\). Here \(d _{i }(S)\) denotes the length of a longest segment in \(S\) parallel to the \(i\)th coordinate axis. It was called axial diameter by \textit{P. R. Scott} [Q. J. Math., Oxf. II. Ser. 36, 359--362 (1985; Zbl 0573.52017)]. More explicitly \(\sigma\) may be expressed as a sum of the absolute values of the entries of \(A^{-1}\), where \(A\) is a matrix built from the coordinates of the vertices of \(S\).