Sums of distances in normed spaces (Q1804728)

For a finite subset \(\{x_1,\dots,x_r\}\) of a real normed space \(X\), put \(s(x_1,\dots,x_r)= \sum|x_i-x_j|\), where the sum is taken over all integers \(i\), \(j\), \(1\leq i<j\leq r\). Using some inequalities from the theory of autonomous systems of differential equations, \textit{M. Martelli} and \textit{S. Busenberg} proved the inequality: (1) \(s(x_1,\dots,x_r)\geq 2(r-1)(1-d)\), where \(d\) denotes the distance from the origin to the convex hull \(C\) of the points \(\{x_i\}\) [Proc. Int. Conf., Columbus/OH (USA) 1988, 183-188 (1989; Zbl 0723.46007)]. The author shows that (1) can be obtained from the inequality: (2) \(s(x_1,\dots,x_r)\geq 2(r-1)\min|x_i-p|\), where \(p\) is a point in \(C\). The inequality (2) was conjectured by B. Grünbaum and proved by the author and \textit{A. D. Andrew} in [Congr. Numerantium 50, 31-35 (1985; Zbl 0592.52006)]. The paper contains also a brief survey of various results related to the inequality (1) in Minkowski spaces, based on methods of integral geometry in the Euclidean space.