Consistent Conical Bicombings in Metric Spaces

Definitions and mathematical statement

Let $(X,d)$ be a metric space. A geodesic bicombing on $X$ is a map \[ \sigma:X\times X\times[0,1]\longrightarrow X \] such that, for every $x,y\in X$ and $s,t\in[0,1]$, \[ \sigma(x,y,0)=x,\qquad \sigma(x,y,1)=y,\qquad d\left(\sigma(x,y,s),\sigma(x,y,t)\right)=|s-t|\,d(x,y). \] It is conical if, for every $x,y,x',y'\in X$ and $t\in[0,1]$, \[ d\left(\sigma(x,y,t),\sigma(x',y',t)\right) \leq (1-t)d(x,x')+t\,d(y,y'). \] It is consistent if, for every $x,y\in X$, $0\leq a\leq b\leq1$, and $u\in[0,1]$, \[ \sigma\left(\sigma(x,y,a),\sigma(x,y,b),u\right) =\sigma\left(x,y,(1-u)a+ub\right). \] These definitions use directed segments; reversibility, the additional identity $\sigma(y,x,t)=\sigma(x,y,1-t)$, is not imposed. The supplied record does not specify whether its terminology includes reversibility, so that convention requires source confirmation.

Question as recorded. Is it true that, for every metric space $(X,d)$, the existence of a conical geodesic bicombing on $X$ implies the existence of a consistent conical geodesic bicombing on $X$? The latter bicombing may differ from the initially given one.

Unresolved scope qualification. The recorded question imposes no completeness hypothesis, whereas the supplied literature assessment describes the unresolved general assertion for complete metric spaces. A metric space is complete if every Cauchy sequence converges in it. The record does not resolve whether completeness was intended as a hypothesis; the question above retains the literal quantification over all metric spaces.

The supplied dated curation labels the problem open, reports that the proper-space case is settled under unspecified weaker convexity assumptions, and reports no known counterexample to the complete-space assertion. Here proper means that every closed bounded subset is compact. These are supplied literature assessments, not independently verified mathematical claims; the unspecified weaker assumptions do not determine a precise auxiliary theorem.

Short English statement

If a metric space admits a conically controlled choice of geodesic between every pair of points, must it admit such a choice that agrees with its own subsegments? The record leaves completeness and the convention concerning reversibility unresolved. Its open-status assessment is supplied dated curation.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30004730$, title Consistent Conical Bicombings in Metric Spaces, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2021$ and names no proposer. Its supplied status is open; its statement status is corrected_verified, with a supplied claim of manual semantic, typographic, and LaTeX review. These labels and that review claim have not been independently verified here.
• Supplied report citation: Differentialgeometrie im Grossen (2022), Oberwolfach Reports, pp. 1685--1734, DOI 10.4171/owr/2021/32. https://doi.org/10.4171/owr/2021/32. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation's parenthetical year $2022$ differs from the DOI's year component and the recorded proposed year $2021$; this discrepancy is unresolved.
• Supplied literature source: "Basso, conical bicombings." https://ems.press/journals/lem/articles/14297776. No fuller bibliographic details are supplied.
• Supplied literature source: "Metric geometry paper on consistent bicombings." https://doi.org/10.1007/s00208-024-02905-w. No authors or precise title are supplied.
• Literature assessment checked at 2026-08-22, exactly as recorded. This is supplied dated curation derived from the supplied OWR curation, not an independently verified assessment. It labels the problem open, reports a result for proper spaces under weaker convexity assumptions, and describes the complete-space assertion as open with no known counterexample. The record itself requests independent verification of time-sensitive status and cited claims.