Geometry of Curves and Surfaces — Problem 3.3

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A convex polyhedron $P$ here is the boundary of the convex hull of finitely many points of $\mathbb R^3$ which do not all lie in a plane. Its edge graph consists of its vertices and edges. A spanning edge tree $T$ contains all vertices and is connected and has no cycles. Cutting along $T$ produces a compact polyhedral disk $P_T$. Its planar development is a map $u_T:P_T\to\mathbb R^2$ that is locally an isometry and preserves lengths on each face. An unfolding is simple if $u_T$ is one-to-one; in that case the developed surface has no overlaps. A polyhedron is unfoldable if some spanning edge tree gives a simple unfolding. The source explains that allowing any collection of edge cuts with connected planar development yields the same question. No affine deformation of $P$ is part of this definition. The surface carries its intrinsic length metric. A pseudo-edge is a distance-minimizing geodesic segment joining two vertices of $P$. A pseudo-edge graph $G$ is a three-connected graph embedded in $P$, with exactly the vertices of $P$, made of pseudo-edges and with convex faces: every face interior angle is less than $\pi$. Three-connected means that deleting any two graph vertices leaves the graph connected. Cutting along a spanning tree $T$ of $G$ gives a surface with a planar development $u_T$. The pair $(P,G)$ is unfoldable if at least one such development is injective.

Statement recorded in the supplied source.

Does there exist a convex polyhedron $P$ with a pseudo-edge graph $G$ such that $(P,G)$ is not unfoldable, so that every spanning-tree cut of $G$ has a noninjective planar development? The cited author paper answers this existence question affirmatively with a 340-vertex example. [S2]

Short English statement

Can shortest geodesic edges form a graph on a convex polyhedron for which every spanning-tree unfolding overlaps? The source constructs such an example.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7000016 (AMR-069-0016). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Nicholas Barvinok and Mohammad Ghomi, Pseudo-edge unfoldings of convex polyhedra, arXiv:1709.04944v4 (2019), Introduction, pp. 1--2, including Theorem 1.1; Section 3.1, pp. 4--5. https://arxiv.org/pdf/1709.04944v4.