Branching Complexity in the Branched Transportation Mailing Problem

Definitions and mathematical statement

Let $\mu^-$ and $\mu^+$ denote the initial and target measures, respectively, and write \[ S=\operatorname{supp}(\mu^-)\cup\operatorname{supp}(\mu^+). \] Here the support of a measure is the set of points every open neighbourhood of which has positive measure. Let $T$ be an optimizer of the mailing problem in branched transportation referred to in the supplied source.

The source asks whether both of the following properties hold:

1. In every ball $B$ outside the supports of the initial and target measures, the optimizer $T$ has finite graph structure. Reading "outside" as disjointness gives the condition $B\cap S=\varnothing$.
2. The number of branches incident to a branching point is uniformly bounded. Writing $\deg_T(v)$ for this number, the requested bound has the form \[ \deg_T(v)\leq N<\infty \qquad\text{for every branching point }v, \] with one bound $N$ over the intended range of branching points and optimizers.

The supplied record does not define the mailing problem's ambient space, admissible measures and traffic plans, prescribed transport data, cost functional, or parameter range. It also does not specify the geometric set associated with $T$ on which graph structure and branching are to be measured. Consequently, "finite graph structure" and $\deg_T(v)$ remain source terms requiring definitions: no choice between topological and geometric graph structure, or between local branches and multiplicity, is imposed here. The record does not resolve whether the balls themselves or their closures must avoid $S$, whether an additional compactness condition on the balls is intended, or what data the bound $N$ may depend on. These missing qualifications prevent a fully specified mathematical formulation; the displayed conditions record the supplied target subject to those unresolved choices.

Short English statement

The question asks whether an optimal mailing network has finite graph structure in every ball away from the initial and target supports, and whether its branching degrees admit a uniform finite bound. The supplied record leaves the optimization model, graph interpretation, and scope of uniformity unspecified. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004485$, title "Branching Complexity in the Branched Transportation Mailing Problem," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection.
• Supplied report citation: Calculus of Variations (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/22, pp. 1139--1196. Supplied source link: https://doi.org/10.4171/owr/2020/22. The registry gives proposed year $2020$; no individual proposer is supplied.
• Supplied literature source: "OWR Calculus of Variations report," https://ems.press/content/serial-article-files/46860.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, states that the cited research context records it as open, and reports that no general uniform bound was found. This is supplied dated curation, not an independently verified literature assessment. The record expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting that it is self-contained and preserves the mathematical target. These are supplied provenance claims; the missing mathematical specifications identified here remain unresolved, and no source visit or independent verification is asserted.