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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (4)

Label	Occurrences
Carathéodory number	1
Carathéodory theorem	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’s theorem in convex geometry self-linksurface differs ⓘ surface form: Carathéodory number
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	Carathéodory’s theorem in convex geometry self-linksurface differs ⓘ surface form: colorful Carathéodory theorem
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

Referenced by (4)

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

Constantin Carathéodory → notableWork → Carathéodory’s theorem in convex geometry ⓘ

Riemann mapping theorem → relatedTo → Carathéodory’s theorem in convex geometry ⓘ

this entity surface form: Carathéodory theorem

Carathéodory’s theorem in convex geometry → concept → Carathéodory’s theorem in convex geometry self-linksurface differs ⓘ

this entity surface form: Carathéodory number

Carathéodory’s theorem in convex geometry → strengthenedBy → Carathéodory’s theorem in convex geometry self-linksurface differs ⓘ

this entity surface form: colorful Carathéodory theorem

All labels observed (4)

How this entity was disambiguated Show

Statements (45)

How these facts were elicited Show

Referenced by (4)

How this entity was disambiguated

How these facts were elicited