Radon’s theorem
E506848
Radon’s theorem is a fundamental result in convex geometry stating that any set of sufficiently many points in Euclidean space can be partitioned into two disjoint subsets whose convex hulls intersect.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Radon’s theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5256294 — 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: Radon’s theorem Context triple: [Carathéodory’s theorem in convex geometry, relatedTo, Radon’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.
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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.
de Bruijn–Erdős theorem
The de Bruijn–Erdős theorem is a fundamental result in combinatorics and graph theory that relates finite and infinite structures, notably asserting that certain properties of infinite graphs or set systems are determined by their finite substructures.
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E.
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.
- 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: Radon’s theorem Target entity description: Radon’s theorem is a fundamental result in convex geometry stating that any set of sufficiently many points in Euclidean space can be partitioned into two disjoint subsets whose convex hulls intersect.
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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.
-
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.
de Bruijn–Erdős theorem
The de Bruijn–Erdős theorem is a fundamental result in combinatorics and graph theory that relates finite and infinite structures, notably asserting that certain properties of infinite graphs or set systems are determined by their finite substructures.
-
E.
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.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
result in convex geometry
ⓘ
theorem ⓘ |
| appliesTo |
Euclidean space R^d
ⓘ
finite sets of points in Euclidean space ⓘ |
| assumes | finite-dimensional Euclidean space ⓘ |
| category |
theorems in convex geometry
ⓘ
theorems in discrete geometry ⓘ |
| consequence |
existence of a point in the intersection of two convex hulls
ⓘ
structure of convex sets in finite-dimensional spaces ⓘ |
| coreIdea | among d+2 points in R^d there is a nontrivial affine dependence with coefficients of both signs ⓘ |
| defines | Radon number of a space ⓘ |
| dimensionParameter | d ⓘ |
| equivalentTo | statement that any d+2 points in R^d are affinely dependent ⓘ |
| field |
combinatorial geometry
ⓘ
convex geometry ⓘ discrete geometry ⓘ |
| generalizedBy |
Tverberg’s theorem
NERFINISHED
ⓘ
topological Radon theorem NERFINISHED ⓘ |
| guarantees |
existence of a Radon partition
ⓘ
intersection of convex hulls of two disjoint subsets ⓘ |
| hasGeneralization |
colorful Radon theorem
NERFINISHED
ⓘ
fractional Helly-type results ⓘ |
| hasVersion |
finite-dimensional version
ⓘ
topological version ⓘ |
| holdsFor | real affine spaces ⓘ |
| implies |
Carathéodory’s theorem
NERFINISHED
ⓘ
Helly’s theorem NERFINISHED ⓘ |
| inspired | study of Radon numbers in abstract convexity ⓘ |
| involvesConcept |
Radon partition
ⓘ
Radon point NERFINISHED ⓘ affine dependence ⓘ convex hull ⓘ |
| minimumNumberOfPoints | d+2 ⓘ |
| namedAfter | Johann Radon NERFINISHED ⓘ |
| originalAuthor | Johann Radon NERFINISHED ⓘ |
| relatedTo |
Erdős–Szekeres-type results in discrete geometry
ⓘ
centerpoint theorem ⓘ |
| requires | at least d+2 points in R^d ⓘ |
| statement | Every set of d+2 points in R^d can be partitioned into two disjoint subsets whose convex hulls intersect. ⓘ |
| typicalProofMethod |
affine dependence arguments
ⓘ
induction on dimension ⓘ linear algebra ⓘ |
| usedIn |
combinatorial optimization
ⓘ
computational geometry ⓘ theory of convex polytopes ⓘ |
| usedInProofOf |
Carathéodory’s theorem
NERFINISHED
ⓘ
Helly’s theorem ⓘ |
| yearProvedApprox | 1921 ⓘ |
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: Radon’s theorem Description of subject: Radon’s theorem is a fundamental result in convex geometry stating that any set of sufficiently many points in Euclidean space can be partitioned into two disjoint subsets whose convex hulls intersect.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.