Carathéodory’s theorem in convex geometry

E118706

result in convex analysis 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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All labels observed (6)

Label	Occurrences
Carathéodory’s theorem	3
Carathéodory theorem	2
Carathéodory's theorem	2
Carathéodory number	1
Carathéodory’s theorem in convex geometry canonical	1
colorful Carathéodory theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T998592 — 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: Carathéodory’s theorem in convex geometry
Context triple: [Constantin Carathéodory, notableWork, Carathéodory’s theorem in convex geometry]

A. Euler’s polyhedron formula
Euler’s polyhedron formula is a fundamental result in topology and geometry that relates the numbers of vertices, edges, and faces of a convex polyhedron through the equation V − E + F = 2.
B. 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.
C. Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
D. Israel–Carter–Robinson uniqueness theorems
The Israel–Carter–Robinson uniqueness theorems are a set of results in general relativity showing that stationary, asymptotically flat black holes in four-dimensional spacetime are completely characterized by just their mass, charge, and angular momentum.
E. Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
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: Carathéodory’s theorem in convex geometry
Target entity description: 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.

A. Euler’s polyhedron formula
Euler’s polyhedron formula is a fundamental result in topology and geometry that relates the numbers of vertices, edges, and faces of a convex polyhedron through the equation V − E + F = 2.
B. 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.
C. Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
D. Israel–Carter–Robinson uniqueness theorems
The Israel–Carter–Robinson uniqueness theorems are a set of results in general relativity showing that stationary, asymptotically flat black holes in four-dimensional spacetime are completely characterized by just their mass, charge, and angular momentum.
E. Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
F. None of above. chosen

Statements (45)

Predicate	Object
instanceOf	result in convex analysis ⓘ theorem in convex geometry ⓘ
appliesTo	points in the convex hull of a set ⓘ subsets of ℝⁿ ⓘ
assumes	standard Euclidean structure on ℝⁿ ⓘ
assumption	ambient space is finite-dimensional ⓘ
bounds	the number of points needed in a convex combination representing a convex-hull point ⓘ
category	theorem about convex hull representations ⓘ
concept	Carathéodory number ⓘ linked to: Carathéodory’s theorem in convex geometry
consequence	convex hull of a set in ℝⁿ equals the set of convex combinations of at most n+1 points from the set ⓘ
domain	Euclidean space ℝⁿ ⓘ
equivalentTo	the statement that the Carathéodory number of ℝⁿ equals n+1 ⓘ
field	convex analysis ⓘ convex geometry ⓘ discrete geometry ⓘ
generalizationOf	the fact that in ℝ² any point in a convex hull is a convex combination of at most 3 points ⓘ the fact that in ℝ³ any point in a convex hull is a convex combination of at most 4 points ⓘ
guarantees	existence of a representation of a convex-hull point using at most n+1 points ⓘ
historicalPeriod	early 20th century mathematics ⓘ
holdsIn	any real finite-dimensional normed vector space (via linear isomorphism with ℝⁿ) ⓘ
implies	every point in the convex hull of a finite set in ℝⁿ is a convex combination of at most n+1 of its points ⓘ
inspired	various colorful and fractional Helly-type theorems ⓘ
involves	affine independence ⓘ barycentric coordinates ⓘ
mathematicalSubjectClassification	52A20 ⓘ
namedAfter	Constantin Carathéodory ⓘ
proofTechnique	induction on the dimension n ⓘ use of affine dependence and Radon partitions ⓘ
relatedTo	Helly’s theorem ⓘ Krein–Milman theorem ⓘ Minkowski’s theorem on convex sets ⓘ Radon’s theorem ⓘ
requires	nonempty subset of ℝⁿ ⓘ point belonging to the convex hull of the subset ⓘ
statement	Any point in the convex hull of a subset of ℝⁿ can be written as a convex combination of at most n+1 points of that subset. ⓘ
strengthenedBy	colorful Carathéodory theorem ⓘ linked to: Carathéodory’s theorem in convex geometry
topic	convex combination ⓘ convex hull ⓘ finite-dimensional vector spaces ⓘ
typeOf	finiteness theorem for convex hull representations ⓘ
upperBound	n+1 points in ℝⁿ ⓘ
usedIn	combinatorial geometry ⓘ computational geometry ⓘ linear programming theory ⓘ optimization ⓘ

How these facts were elicited