On averaging multisets (Q1909220)

The path rank of \(M \subseteq \mathbb{R}^s\), denoted \(\text{prk }M\), is defined as the maximum number of (continuous) paths \(\varphi : [0,1] \to M\) whose principal chords \(\varphi(1) - \varphi(0)\) are independent. The author proves that if \(\text{prk } M = s\) then each point in the interior of the convex hull of \(M\) can be written as a convex combination with rational coefficients of \(2s\) points in \(M\). This bound is sharp, as shown by an example. However, for \(s > 1\), if for each \(m \in M\) there exists a nonconstant path \(\varphi : [0,1] \to M\) such that \(\varphi(0) = m\), the bound \(2s\) can be lowered to \(2s-1\). This bound is sharp as well.