Alexandrov's theorem on polyhedra

Every metric on a topological sphere formed by gluing finitely many Euclidean polygons, with positive curvature concentrated at vertices, is realized by a convex polyhedron. That polyhedron is unique up to rigid motion.

Connect

Polyhedral metric: A metric that is locally Euclidean except at finitely many points, where the total angle may differ from 2π. The theorem takes this intrinsic surface geometry as its input.
Pogorelov's proof of Alexandrov's theorem: A proof establishing existence and rigidity for convex polyhedra with prescribed intrinsic metrics. Pogorelov developed a central proof of the theorem's realization and uniqueness claims.
Alexandr Danilovich Alexandrov: A Soviet mathematician whose work shaped modern convex geometry and the theory of intrinsic metrics. He proved the realization theorem and developed its geometric setting.
Discrete differential geometry: The study of discrete structures that parallel ideas from differential geometry. Polyhedral metrics are discrete geometric data that this field analyzes and reconstructs.
Gaussian curvature: A measure of intrinsic curvature at a point on a surface, defined locally by how distances behave. Its concentrated, positive form at vertices is the theorem's curvature condition.
Alexandrov gluing theorem: A theorem characterizing metrics obtained by gluing Euclidean polygons into convex polyhedral surfaces. It provides the metric-side characterization underlying Alexandrov's realization result.
Convex geometry: The study of convex sets and their geometric properties. Alexandrov's theorem belongs to a wider effort to characterize convex bodies.
Polyhedral reconstruction: The recovery of a polyhedral surface from specified geometric measurements or constraints. Alexandrov's theorem guarantees a convex realization from intrinsic metric data.
Angle defect: At a polyhedral vertex, 2π minus the sum of incident face angles. Positive angle defects express positive curvature at the metric's vertices.
Curvature measure: A measure that records curvature, including curvature concentrated at singular points. For polyhedral metrics, the vertex angle defects form the curvature data used in the construction.