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 (1)

Label	Occurrences
Radon’s theorem canonical	1

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 ⓘ

Referenced by (1)

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

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