Metric Diameters of Measures with Prescribed Interval Masses

Definitions and mathematical statement

Let $h>0$, and let $(a_n)_{n\geq1}$ be the prescribed sequence of nonnegative masses in the supplied record. Retaining its notation, set \[ A(h,(a_n))=\left\{\mu\in M_+({\mathbb{R}}_+):\ \mu([nh,(n+1)h])=a_n\ \text{for every integer }n\geq1\right\}. \] Here $M_+({\mathbb{R}}_+)$ denotes the source's class of nonnegative measures on ${\mathbb{R}}_+$; the supplied record does not specify the measurable structure, finiteness or local finiteness requirements, or the convention for ${\mathbb{R}}_+$.

For a nonempty set $B$ on which a distance $\rho$ is defined, its diameter is \[ \operatorname{diam}_{\rho}(B)=\sup\{\rho(\mu,\nu):\mu,\nu\in B\}, \] allowing the supremum to be $+\infty$. The requested problem is to determine \[ \operatorname{diam}_{\rho_F}\left(A(h,(a_n))\right) \quad\text{and}\quad \operatorname{diam}_{\rho_W}\left(A(h,(a_n))\right), \] and describe their dependence on $h$, with the prescribed sequence held fixed. The record calls $\rho_F$ the flat metric and $\rho_W$ a Wasserstein-type (also described as Wasserstein-like) metric, but supplies neither definition. Consequently, these expressions retain the source's symbols and do not specify unique distances without further information.

The intervals are closed exactly as supplied: adjacent intervals share endpoints, so an atom at a shared endpoint contributes to both corresponding constraints. The indexing begins at $n=1$; no interval-mass condition is imposed on ${\mathbb{R}}_+\cap[0,h)$. No summability, total-mass, moment, or nonemptiness hypothesis is supplied. These omissions must be resolved, together with the metric definitions and their domains, before the diameter questions have a fully specified interpretation; no convention for an empty-set diameter is imposed here.

Short English statement

Given a positive interval length $h$ and prescribed nonnegative masses $a_n$, consider all measures assigning mass $a_n$ to each closed interval $[nh,(n+1)h]$, for every $n\geq1$. Determine the diameter of this family in the source's flat and Wasserstein-type metrics and how each diameter depends on $h$. The metric definitions, measure class, and necessary admissibility hypotheses are missing from the supplied record, so this remains a qualified statement requiring mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete source record, permanent ID $30001252$, source problem code catalogue entry, title: Metric Diameters of Measures with Prescribed Interval Masses. The registry assigns proposed year 2009 and supplies no named proposer.
• Supplied report citation: Mathematical Biology (2009), Oberwolfach Reports, pp. 1303--1374, DOI 10.4171/owr/2009/24. Source URL and supplied literature-source link, labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2009/24. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page or metric definitions are supplied.
• Supplied literature assessment, checked on 2026-08-21: the registry labels the problem open and states that the report gives upper bounds for several metrics while leaving the exact diameter and its dependence on $h$ undetermined. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.