Length-Convex Hulls in Teichmüller Space

Definitions and mathematical statement

The supplied question concerns a Teichmüller space $\mathcal T$, a family $\Gamma$ of curves, and curve-length functions $\ell_\gamma$ on $\mathcal T$. The underlying surface, the permitted curves, the meaning of length, and the precise convention for length-convexity are not specified in the supplied record.

The following gives a conditional formulation with these choices exposed. Fix a surface $S$, its intended Teichmüller space $\mathcal T(S)$, a curve family $\Gamma$, and functions $\ell_\gamma:\mathcal T(S)\to{\mathbb{R}}$ for $\gamma\in\Gamma$. If length-convex sets mean arbitrary intersections of closed sublevel sets, put \[ B_{\gamma,t}=\{X\in\mathcal T(S):\ell_\gamma(X)\le t\},\qquad \gamma\in\Gamma,\ t\in{\mathbb{R}}, \] and define \[ \mathcal C_\ell=\left\{\bigcap_{B\in\mathcal B}B:\mathcal B\subseteq\{B_{\gamma,t}:\gamma\in\Gamma,\ t\in{\mathbb{R}}\}\right\}, \] where the empty intersection is $\mathcal T(S)$. For any $p,q\in\mathcal T(S)$, the associated length-convex hull is, by definition, \[ \operatorname{Hull}_\ell\{p,q\} =\bigcap\{C\in\mathcal C_\ell:p,q\in C\}. \] Question. Determine the length-convex hull of $\{p,q\}$ for arbitrary $p,q$ in the intended Teichmüller space.

The displayed definitions are conditional on the closed-sublevel and arbitrary-intersection interpretation; the supplied record does not establish that this is the exact source convention. They specify the object under that interpretation and do not constitute a determination of the requested hull. No assumption that $p\ne q$ is supplied. The background also mentions an extremal-length analogue, but does not state its definitions or a separate precise question; that contextual qualification remains unresolved.

Short English statement

Describe the hull of two arbitrary points in Teichmüller space for the notion of convexity generated by curve-length sublevel sets. The surface, curve family, length functions, and sublevel-set convention require clarification. The supplied background also mentions an extremal-length analogue without formulating it precisely.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30001638$, source problem code catalogue entry, entitled "Length-Convex Hulls in Teichmüller Space." The recorded proposed year is 2010; no individual proposer is supplied.
• Supplied source citation: Teichmüller Theory (2011), Oberwolfach Reports, pp. 3085--3157, DOI 10.4171/owr/2010/53. Original Oberwolfach problem report: https://doi.org/10.4171/owr/2010/53. This is the authoritative source identified by the supplied record; it has not been independently consulted for this draft.
• Supplied related literature reference, labelled "convex hulls in thick Teichmuller space": https://doi.org/10.1007/s11425-014-4871-6. No author information or fuller bibliographic citation is supplied.
• Supplied literature assessment dated 2026-08-21: the recorded status is open. The assessment says that related metric-convexity questions have counterexamples, while that literature does not resolve the requested two-point length-convex hull or its extremal-length analogue. This is supplied dated curation derived from the OWR curation, not an independently verified assessment; time-sensitive status and cited claims require independent verification.
• The supplied statement status is corrected_verified. Its accompanying verification note reports that the mathematical content was checked against the official Oberwolfach report and that notation, context, and question form were normalized without changing the source-level claim. This describes supplied provenance, not verification performed for this draft.