Alexandrov's Theorem on Polyhedra is a rigidity theorem describing three-dimensional convex polyhedra in terms of the distances between points measured along their own surfaces. It shows that convex polyhedra of different shapes always have distinct surface metrics, and it characterizes exactly which abstract metric spaces arise as the surface-distance geometry of some convex polyhedron. Named for the Soviet mathematician Aleksandr Danilovich Aleksandrov, who published it in the 1940s, it is a foundational result connecting convex geometry to metric geometry.
Facts
Partially Attested
Proof Year1942 is the year of the detailed Matematicheskii Sbornik paper; the record does not say whether it is the first publication, and Wikipedia gives only the 1940s. StatementA metric space that is geodesic, homeomorphic to a sphere, and locally Euclidean except for a finite number of cone points of positive angular defect is the surface metric of a convex polyhedron, which is unique up to congruence. 1 Classification
Statement Form Connections
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Alexandrov's theorem on polyhedra (Wikipedia)
Statement of the theorem, final paragraphQuote, Statement of the theorem, final paragraph
there exists a convex polyhedron whose development is the given space
2. A. Alexandroff, Existence of a convex polyhedron and of a convex surface with a given metric, Mat. Sbornik 1942 (Math-Net.Ru)
Bibliographic recordQuote, Bibliographic record
Rec. Math. [Mat. Sbornik] N.S., 11(53):1-2 (1942)
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