Rearrangement of vector series. II (Q2709859)

This paper is a continuation of the theory in the first part (see Zbl 0983.40001)]. For each permutation \(f\) of the positive integers, the authors use the max-flow min-cut theorem of graph theory to determine all convex sets in \(C_f(\mathbb{R}^d)\) which are symmetric about a point. These sets depend only on a parameter \(w(f)\in \mathbb{N}\cup \{0,+ \infty\}\), called the width of \(f\). It is shown that \(w(f)\), when it is a positive integer, falls far short of completely determining \(C_f(\mathbb{R}^d)\), but, for each \(q\in\mathbb{N}\), we can find the largest of the sets in \(C_f(\mathbb{R}^d)\) arising from permutations \(f\) of width \(q\). The smallest of these sets when \(q=1\) is also described.