Recovering from some surgeries, I wanted to write a little something on Stoker’s conjecture: It is best viewed as an angular converse to Cauchy’s rigidity theorem. Cauchy’s proves more than the usual statement that congruent corresponding faces determine a convex polyhedron: for a fixed combinatorial type, the angles inside the faces already determine the dihedral angles. In 1968, James J. Stoker asked whether the converse holds: do the dihedral angles determine the face angles?
This is a nice conjecture, and has attracted some attention over the decades. Note that the dihedral angles cannot determine the polyhedron fully. One can only hope that the polyhedron is determined up to normal equivalence, that is, up to a movement of the defining hyperplanes in their normal directions.
Still, there was immediate evidence for the conjecture. Stoker’s fifty-page article was followed in the same 1968 issue of Communications on Pure and Applied Mathematics by a six-page paper of Hermann Karcher, who verified the conjecture for a class of convex polyhedra with five vertices and six triangular faces. But the general problem proved unexpectedly resistant. Related geometries gave a mixed picture: Andreev’s theorem established the analogous rigidity for non-obtuse convex hyperbolic polyhedra, while Jean-Marc Schlenker later showed that the spherical analogue is false.
A major step came from the deformation theory of cone-manifolds. Given a convex polyhedron, one can double it across its faces; its edges then become the singular locus of a Euclidean or hyperbolic cone-manifold, and the cone angles are twice the dihedral angles. Using this correspondence and elliptic analysis on singular spaces, Rafe Mazzeo and Grégoire Montcouquiol proved the infinitesimal Stoker conjecture: an infinitesimal deformation of a convex Euclidean polyhedron that preserves all dihedral angles must also preserve all face angles. This was a strong local rigidity statement, but it did not yet compare two polyhedra that might be far apart in their moduli space.
The first complete global proof was given by Jinmin Wang and Zhizhang Xie in 2022, with a revised version appearing in 2023. They proved Stoker’s conjecture in every dimension as a consequence of a much broader theorem: Gromov’s flat-corner domination conjecture. Their method uses index theory for Dirac operators on manifolds with polyhedral boundary and is designed to compare scalar curvature, boundary mean curvature, and dihedral angles. It is powerful, but Stoker’s conjecture appears there as one consequence of a substantial analytic and index-theoretic framework. And it relies on an earlier work that had to undergo several corrections, and counterexamples to earlier versions of that work.
A new paper by Yuchen Bi’s paper opened a shorter route. Building on Brendle’s smoothing method, Bi replaces a polytope by smooth convex inner approximations, constructs an approximate Gauss map on their boundaries, and applies a Dirac boundary-value argument before passing to a flat limiting spinor. The present note adapts that construction to a relative problem involving two polytopes. Instead of following the normals of one polytope, the approximate normal map is made to travel between the corresponding normals of the other. This proves a one-sided strengthening of Stoker’s conjecture: if all corresponding exterior dihedral angles are ordered in the appropriate direction, then equality is forced and the two complete systems of facet normals differ by a single orthogonal transformation. Thus the entire spherical normal fan, not merely the face angles, is determined.
The resulting proof is not elementary—it still uses harmonic spinors and the Dirac operator—but its polyhedral part is comparatively concrete: smooth the boundary, control how the normal rotates near each ridge, patch the construction near the codimension-three skeleton, and extract the normal fan from the limiting Clifford relations. A genuinely polyhedral or combinatorial proof remains open. You can grab the note here.
Unsolicited Nonsense 