Open problems from the Length Optimization event at MDS 2026

Abstract

During the Minisymposium on Length Optimization as part of the International Conference "Mathematics Days in Sofia - 2026", the participants collected the following open problems: 1. Find any non-trivial upper and lower bounds for the degree of a branching point in the Gilbert-Steiner problem with cost-function $c(x) = x^p$ in the metric space $X = \mathbb{R}^d$ for large $d$ and fixed $p > 1/2$ (see Section 1). 2. Determine whether a typical trajectory of a given branched optimal traffic plan has bounded length. To find the optimal transport plan between the Dirac measure in the center of a circle and the Lebesgue measure on the circle (see Section 2). 3. To find the maximal distance minimizers for any explicit compact sets (see Section 3). 4. What is the minimal $C$ such that for any set of $n$ points in the three-dimensional unit cube, there is a tour through all these $n$ points with the sum of the cubes of its edge lengths at most $C$ (see Section 4). 5. To explore parallels between tropical caustics and Steiner trees: do caustics really need convexity (see Section 5).