Helly’s theorem
E506847
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.
All labels observed (7)
| Label | Occurrences |
|---|---|
| Helly's theorem | 2 |
| Helly’s theorem canonical | 2 |
| Helly theorem | 1 |
| Helly-type theorems | 1 |
| Helly-type theorems for algebraic sets | 1 |
| quantitative Helly theorem | 1 |
| topological Helly theorem | 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]
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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.
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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.
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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.
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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 ⓘ |
| 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
ⓘ
linked to:
Helly’s theorem
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
ⓘ
quantitative Helly theorem ⓘ
linked to:
Helly’s theorem
topological Helly theorem ⓘ
linked to:
Helly’s theorem
|
| 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 ⓘ |
| originallyProvedBy | Eduard Helly ⓘ |
| publicationLanguage | German ⓘ |
| relatedTo |
(p,q)-theorem
ⓘ
Carathéodory’s theorem ⓘ
linked to:
Carathéodory’s theorem in convex geometry
Radon’s theorem ⓘ Tverberg’s theorem ⓘ colorful Helly theorem ⓘ fractional Helly theorem ⓘ |
| specialCaseOf |
Helly-type theorems
ⓘ
linked to:
Helly’s theorem
|
| usedIn |
computational geometry
ⓘ
discrete optimization ⓘ geometric algorithms ⓘ geometric transversal theory ⓘ linear programming theory ⓘ optimization ⓘ |
| yearProvedApprox | 1913 ⓘ |
| yearPublishedApprox | 1923 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Helly’s theorem Description of subject: 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.
Referenced by (9)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Helly’s theorem
linked to: Helly’s theorem
linked to: Helly’s theorem
linked to: Helly’s theorem
linked to: Helly’s theorem
linked to: Helly’s theorem
linked to: Helly’s theorem