Helly's Theorem states that for a finite collection of convex sets in d-dimensional space, if every subcollection of d plus one of the sets has a point in common, then the entire collection has a point in common. Named for Eduard Helly, it is a foundational result of convex geometry that reduces checking a common intersection for arbitrarily many convex sets to checking only small subcollections of a bounded size, and it underlies later results such as Radon's Theorem and the centerpoint theorem.
Facts
StatementFor a finite collection of convex subsets of R^d with n >= d+1, if the intersection of every d+1 of them is nonempty, then the whole collection has a nonempty intersection. 1 Classification
Statement Form Connections
Has Statement Form
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.
In Branch
Source Helly's theorem and related results (Wikipedia)
Sources
1. Helly's theorem and related results (Wikipedia)
Statement
Let X1, ..., Xn be a finite collection of convex subsets of R d, with n ≥ d + 1. If the intersection of every d + 1 of these sets is nonempty, then the whole collection has a nonempty intersection.
Introduction, discovery
It was discovered by Eduard Helly in 1913, but not published by him until 1923, by which time alternative proofs by Radon (1921) and König (1922) had already appeared.
In Branch: Discrete Geometry, Lead sentence
Helly's theorem is a basic result in discrete geometry on the intersection of convex sets.
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