Bipolar Comparison and Transport Continuity

Definitions and mathematical statement

For a metric space $(X,d)$, the standard metric formulation of $(4,1)$-bipolar comparison is the following condition: for every choice of points \[ a_0,a_1,a_2,a_3,a_4,b_0,b_1\in X, \] there exist corresponding points $\widehat a_0,\ldots,\widehat a_4,\widehat b_0,\widehat b_1$ in a real Hilbert space $H$ such that \[ \begin{aligned} \|\widehat a_i-\widehat a_0\|_H&=d(a_i,a_0) &&(1\leq i\leq4),\\ \|\widehat b_1-\widehat b_0\|_H&=d(b_1,b_0),\\ \|\widehat a_0-\widehat b_0\|_H&=d(a_0,b_0), \end{aligned} \] and \[ \|\widehat u-\widehat v\|_H\geq d(u,v) \] for every pair of labeled points $u,v$ in this configuration. Write $B_{4,1}(X)$ for this condition. This standard formulation is included for clarity; its agreement with the intended source convention requires review because the supplied record does not give a definition.

Let $\mathcal C$ denote the intended class of spaces, with their relevant metrics and any additional transport structure, and let $\mathsf{TCP}(X)$ denote the transport continuity property intended by the source. The supplied record specifies neither $\mathcal C$ nor a definition of $\mathsf{TCP}$. In particular, it does not identify the transport cost, the admissible measures, the existence or uniqueness requirements on transport maps, or the precise continuity requirement.

Question, conditional on these missing specifications. Is \[ \forall X\in\mathcal C,\qquad B_{4,1}(X)\Longrightarrow\mathsf{TCP}(X) \] true? More strongly, is \[ \forall X\in\mathcal C,\qquad B_{4,1}(X)\Longleftrightarrow\mathsf{TCP}(X) \] true? The second question additionally asks for the reverse implication. These are formulation schemas, not fully specified conjectures until the intended class and transport continuity property are supplied. No universal claim over all metric spaces is inferred.

Short English statement

The question asks whether $(4,1)$-bipolar comparison guarantees the transport continuity property, and whether the converse also holds. The intended class of spaces and the exact meaning of transport continuity are missing, so the statement remains conditional on those specifications.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003856$, title Bipolar Comparison and Transport Continuity, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2018 and names no proposer.
• Supplied report citation: Geometrie (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/27, pp. 1635--1684. https://doi.org/10.4171/owr/2018/27. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Lebedeva--Petrunin--Zolotov, Bipolar comparison. https://arxiv.org/abs/1711.09423.
• Supplied literature reference: Oberwolfach Report 27/2018. https://ems.press/content/serial-article-files/46749?nt=1.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, states that Lebedeva--Petrunin--Zolotov relate bipolar comparison to CTIL and MTW, and reports that the implication or equivalence with TCP remains conjectural and that no later primary proof was found. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The supplied background explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, and sentence structure were repaired. This supplied editorial label does not establish mathematical correctness or resolve the missing hypotheses and definitions.