Density Conditions for Optimal-Transport Regularity

Definitions and mathematical statement

The supplied record asks for sufficient density hypotheses for regularity of an optimal transport map when usual geometric hypotheses fail. It does not specify a complete mathematical setting, so the following notation describes the intended question schematically.

Let $X$ and $Y$ denote the unspecified source and target spaces, with reference measures $m_X$ and $m_Y$. Write the source and target measures as \[ {\,\mathrm{d}}\mu=\rho\,{\,\mathrm{d}} m_X,\qquad {\,\mathrm{d}}\nu=\sigma\,{\,\mathrm{d}} m_Y, \] where $\rho$ and $\sigma$ are their nonnegative densities. A measurable transport map $T:X\to Y$ satisfies $T_{\#}\mu=\nu$, meaning \[ \mu(T^{-1}(B))=\nu(B) \] for every measurable target set $B$. Optimality refers to the transport cost and admissible class of the intended problem; neither is specified in the record. The existence or uniqueness of such an optimal map is not asserted here.

Source question. Can hypotheses on $\rho$ and $\sigma$ guarantee regularity of the optimal transport map when standard geometric conditions, such as the Ma--Trudinger--Wang (MTW) condition, fail?

Unresolved scope. The record does not specify the cost, governing equation, dimension, spaces or domains, geometric and boundary hypotheses, density class, or required regularity. It also does not identify the version of the MTW condition or quantify the permitted geometric failures. In particular, it does not distinguish regularity in the interior from regularity up to the boundary, or specify whether a regular representative is sought for an almost-everywhere-defined map. Thus the question is a request to identify sufficient hypotheses in an unspecified setting, rather than a determinate universally quantified conjecture. A precise implication cannot be extracted without additional source information.

Short English statement

Can suitable assumptions on the source and target densities ensure regularity of an optimal transport map when usual geometric assumptions, including an MTW condition, fail? The supplied record leaves both the transport setting and the desired regularity unspecified. Its open label is supplied curation, not an independently verified mathematical status.

Sources

• Local UnsolvedMath registry, permanent ID $30001863$, title Density Conditions for Optimal-Transport Regularity, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2011; no proposer is named.
• Supplied report citation: Partial Differential Equations (2012), Oberwolfach Reports, DOI 10.4171/owr/2011/38, pp. 2163--2215. Supplied source URL and literature-source link, labeled "OWR report 38": https://doi.org/10.4171/owr/2011/38. The background describes the reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page or named contribution is supplied.
• Literature assessment dated 2026-08-21: supplied status open. The supplied assessment says that the extracted question seeks sufficient density conditions for regularity but omits the equation and hypotheses, so no determinate general resolution can be verified. The record lists OWR report 38 as checked. This is supplied dated curation derived from the OWR record, not independently verified source consultation or a new literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review describing the cleanup as self-contained and meaning-preserving. That supplied review claim does not resolve the missing mathematical hypotheses identified above.