Helly’s theorem

E506847

mathematical theorem result in convex geometry

Helly’s theorem is a fundamental result in convex geometry that gives conditions under which a family of convex sets in Euclidean space has a nonempty common intersection.

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All labels observed (1)

Label	Occurrences
Helly’s theorem canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T5256293 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Helly’s theorem
Context triple: [Carathéodory’s theorem in convex geometry, relatedTo, Helly’s theorem]

A. Carathéodory’s theorem in convex geometry
Carathéodory’s theorem in convex geometry is a fundamental result stating that any point in the convex hull of a set in ℝⁿ can be expressed as a convex combination of at most n+1 points from that set.
B. Erdős–Szekeres theorem
The Erdős–Szekeres theorem is a fundamental result in combinatorial geometry that guarantees the existence of large convex polygons within sufficiently large sets of points in the plane in general position.
C. Tucker’s lemma
Tucker’s lemma is a combinatorial analog of the Borsuk–Ulam theorem that provides conditions guaranteeing the existence of certain complementary edge labels in triangulated spheres.
D. Sperner's lemma
Sperner's lemma is a fundamental result in combinatorial topology that guarantees the existence of a fully labeled simplex in certain labeled triangulations, and is widely used to prove fixed-point and equilibrium theorems.
E. Steinhaus theorem
The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Helly’s theorem
Target entity description: Helly’s theorem is a fundamental result in convex geometry that gives conditions under which a family of convex sets in Euclidean space has a nonempty common intersection.

A. Carathéodory’s theorem in convex geometry
Carathéodory’s theorem in convex geometry is a fundamental result stating that any point in the convex hull of a set in ℝⁿ can be expressed as a convex combination of at most n+1 points from that set.
B. Erdős–Szekeres theorem
The Erdős–Szekeres theorem is a fundamental result in combinatorial geometry that guarantees the existence of large convex polygons within sufficiently large sets of points in the plane in general position.
C. Tucker’s lemma
Tucker’s lemma is a combinatorial analog of the Borsuk–Ulam theorem that provides conditions guaranteeing the existence of certain complementary edge labels in triangulated spheres.
D. Sperner's lemma
Sperner's lemma is a fundamental result in combinatorial topology that guarantees the existence of a fully labeled simplex in certain labeled triangulations, and is widely used to prove fixed-point and equilibrium theorems.
E. Steinhaus theorem
The Steinhaus theorem is a fundamental result in measure theory stating that the difference set of any subset of the real numbers with positive Lebesgue measure contains an open interval around zero.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in convex geometry ⓘ
appearsIn	classical convexity theory ⓘ
appliesIn	Euclidean space NERFINISHED ⓘ
appliesTo	families of convex sets ⓘ
assertsThat	for a finite family of convex sets in R^d, if every subfamily of size d+1 has nonempty intersection, then the whole family has nonempty intersection ⓘ
category	theorems in convex analysis ⓘ theorems in geometry ⓘ
coreConcept	intersection of convex sets ⓘ
dimensionDependent	yes ⓘ
field	combinatorial geometry ⓘ convex geometry ⓘ discrete geometry ⓘ
givesConditionFor	nonempty common intersection ⓘ
hasGeneralization	Helly-type theorems for algebraic sets NERFINISHED ⓘ Helly-type theorems for other set systems ⓘ Helly-type theorems in metric spaces ⓘ
hasHellyNumber	d+1 for convex sets in R^d ⓘ
hasParameter	dimension d of Euclidean space ⓘ
hasVariant	Doignon’s theorem NERFINISHED ⓘ quantitative Helly theorem NERFINISHED ⓘ topological Helly theorem NERFINISHED ⓘ
holdsFor	finite families of convex sets ⓘ
implies	finite intersection property for convex sets under Helly’s condition ⓘ
influenced	development of combinatorial convexity ⓘ theory of LP-type problems ⓘ
isFiniteVersionOf	intersection properties of convex sets ⓘ
isToolFor	geometric proofs in combinatorics ⓘ proving existence of feasible solutions in linear inequalities ⓘ
namedAfter	Eduard Helly NERFINISHED ⓘ
originallyProvedBy	Eduard Helly NERFINISHED ⓘ
publicationLanguage	German ⓘ
relatedTo	(p,q)-theorem NERFINISHED ⓘ Carathéodory’s theorem NERFINISHED ⓘ Radon’s theorem NERFINISHED ⓘ Tverberg’s theorem NERFINISHED ⓘ colorful Helly theorem NERFINISHED ⓘ fractional Helly theorem NERFINISHED ⓘ
specialCaseOf	Helly-type theorems NERFINISHED ⓘ
usedIn	computational geometry ⓘ discrete optimization ⓘ geometric algorithms ⓘ geometric transversal theory ⓘ linear programming theory ⓘ optimization ⓘ
yearProvedApprox	1913 ⓘ
yearPublishedApprox	1923 ⓘ

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Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Carathéodory’s theorem in convex geometry → relatedTo → Helly’s theorem ⓘ

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Statements (47)

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Referenced by (1)

How this entity was disambiguated

How these facts were elicited