Continuous Selection of Piecewise Linear Planar Extensions

Definitions and mathematical statement

Let $T=ABC$ denote the fixed reference triangle, interpreted in the usual sense as the closed filled nondegenerate triangle, and let $\partial T$ be its boundary. A map on either of these polyhedra is piecewise linear (PL) if it is continuous and affine on every simplex of some finite triangulation of its domain. The triangulation may depend on the map; the supplied statement imposes no fixed triangulation or bound on its size. Define \[ \mathcal B=\{f:\partial T\to{\mathbb{R}}^2: f\text{ is PL and injective}\},\qquad \mathcal E=\{F:T\to{\mathbb{R}}^2: F\text{ is PL and injective}\}. \] Restriction defines $r:\mathcal E\to\mathcal B$ by $r(F)=F|_{\partial T}$.

The question is whether there exists a continuous selection of extensions, that is, a map \[ S:\mathcal B\longrightarrow\mathcal E, \qquad r\circ S=\operatorname{id}_{\mathcal B}. \] Equivalently, for every $f\in\mathcal B$, the selected map $S(f)$ must be a PL injective map on all of $T$ satisfying $S(f)|_{\partial T}=f$, and the selection must depend continuously on $f$.

Unresolved qualification. The supplied record does not specify topologies on $\mathcal B$ and $\mathcal E$, so continuity of $S$ is not fully defined. One possible precise interpretation equips these spaces with the uniform metrics \[ d_{\partial T}(f,g)=\sup_{x\in\partial T}\|f(x)-g(x)\|_2, \qquad d_T(F,G)=\sup_{x\in T}\|F(x)-G(x)\|_2. \] Under this additional interpretation, the question asks whether such a section $S$ exists with \[ \forall f\in\mathcal B\ \forall\varepsilon>0\ \exists\delta>0\ \forall g\in\mathcal B, \quad d_{\partial T}(f,g)<\delta \ \Longrightarrow\ d_T(S(f),S(g))<\varepsilon. \] This metric formulation is conditional: the supplied evidence does not establish that these are the intended topologies.

Short English statement

Can every injective piecewise linear map of a fixed triangle's boundary into the plane be assigned an injective piecewise linear extension over the whole triangle, with the assignment continuous as the boundary map varies? The source leaves the topologies on the map spaces unspecified; the uniform-topology formulation above therefore requires confirmation.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30004524$, source problem code catalogue entry, title Continuous Selection of Piecewise Linear Planar Extensions, in the Oberwolfach Reports Open Problems collection. The supplied record lists proposed year 2020 and supplies no individual proposer.
• Supplied report citation: Discrete Geometry (2021), Oberwolfach Reports, pp. 1469--1528, DOI 10.4171/owr/2020/30. https://doi.org/10.4171/owr/2020/30. The background describes this collection as drawn from workshop reports and their open-problem discussions.
• Supplied literature source: Rote, Open Problems in Discrete Geometry. https://page.mi.fu-berlin.de/rote/Kram/Problems-Discrete-Geometry-2020.pdf.
• Supplied literature source: Oberwolfach Reports 2020/30. https://ems.press/content/serial-article-files/46867.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the listed collapsibility, centerpoint-constant, two-convex-pieces Helly, and continuous-extension questions remain unresolved in the cited literature. This is supplied dated curation, not an independently verified literature assessment. Its background explicitly requests independent verification of time-sensitive status and cited claims; only the continuous-extension question is formulated here.
• The supplied statement-status label is corrected_verified. Its accompanying qualification says that only formula delimiters, notation, grammar, and typography were repaired, with mathematical content and scope preserved. This supplied editorial label does not establish mathematical correctness or resolve the missing topology.