On the axial diameters of a convex body (Q764046)

Let \(Q_n\) be the unit cube \( [0,1]^n\) and \(S\) a nondegenerate simplex in the space \({\mathbb R}^n\) with a fixed coordinate system. Given a convex body \(C\), let \( \tau C\) be the homothety of \(C\) with respect to its barycenter with homothety ratio \(\tau\), and \( d_i (C) \) be the maximum length of segments contained in \(C\) and parallel to the axis \(Ox_i \). This number is called the \(i\)-th axial diameter of \(C\). The main results of the paper are: Theorem 1. Let \(C\) be a convex body, and let \({\sigma}^{-1}:=\sum_{i=1}^{n} \frac{1}{d_i (C)}.\) Then \(C\) contains a translate of \(\sigma Q_n\). Theorem 2. The inequality \(\sum_{i=1}^{n} \frac{1}{d_i (S)} \leq 1\) holds iff \(Q_n\) lies in a translate of \(S\). Further, \(\sum_{i=1}^{n} \frac{1}{d_i (S)} = 1\) iff \(Q_n\) lies in a translate of \(S\) and each \((n-1)\)-dimensional facet of \(S\) contains a vertex of \(Q_n\). The author also relates this sum with sum of modules of elements of the matrix \({\mathbf A}^{-1}\), where the \(i\)-th line of the \((n+1)\times(n+1) \) matrix \({\mathbf A}\) is composed of coordinates of the \(i\)-th vertex of the simplex \(S\) and the unit in the \((n+1)\)-th column.