Radon’s theorem

E506848

result in convex geometry theorem

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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All labels observed (3)

Label	Occurrences
Radon’s theorem canonical	2
Radon theorem	1
topological Radon theorem	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]

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
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.

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 ⓘ topological Radon theorem ⓘ linked to: Radon’s theorem
guarantees	existence of a Radon partition ⓘ intersection of convex hulls of two disjoint subsets ⓘ
hasGeneralization	colorful Radon theorem ⓘ linked to: colorful Helly theorem fractional Helly-type results ⓘ
hasVersion	finite-dimensional version ⓘ topological version ⓘ
holdsFor	real affine spaces ⓘ
implies	Carathéodory’s theorem ⓘ linked to: Carathéodory’s theorem in convex geometry Helly’s theorem ⓘ
inspired	study of Radon numbers in abstract convexity ⓘ
involvesConcept	Radon partition ⓘ Radon point ⓘ affine dependence ⓘ convex hull ⓘ
minimumNumberOfPoints	d+2 ⓘ
namedAfter	Johann Radon ⓘ
originalAuthor	Johann Radon ⓘ
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 ⓘ linked to: Carathéodory’s theorem in convex geometry Helly’s theorem ⓘ
yearProvedApprox	1921 ⓘ

How these facts were elicited

Referenced by (4)

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

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

Helly’s theorem → relatedTo → Radon’s theorem ⓘ

Radon’s theorem → generalizedBy → topological Radon theorem ⓘ

linked to: Radon’s theorem

Gale transform → relatedTo → Radon theorem ⓘ