Redundance of vertices of the cube relatively to its minimal ellipsoid (Q1922677)

Each convex \(n\)-body \(K \subseteq \mathbb{R}^n\) has a unique ellipsoid of minimal volume containing \(K\). If \(K\) is a polytope, then there exists a unique minimal number \(r(K)\) of vertices of \(K\) whose convex hull has the same minimal ellipsoid as \(K\). The author asks what is the minimal number \(r(Q_n)\) for the \(n\)-dimensional cube \(Q_n\) with the ball \(B_n: = \{x\in \mathbb{R}^n: |x |\leq 1\}\) as minimal ellipsoid. He proves the following interesting theorem: If there exists an Hadamard matrix of order \(m\geq 8\), then holds \(r (Q_n) = m\) for the dimensions \(n = m-4, \dots, m-1\). This supplies \(r(Q_n)\) up to dimension \(n= 423\).