Theorem

Caratheodory's Theorem (Convex Hull)

Geometry

Caratheodory's theorem is a theorem in convex geometry stating that if a point lies in the convex hull of a set of points in d-dimensional space, then that point also lies in some d-dimensional simplex whose vertices are drawn from the same set, meaning it can be written as a convex combination of at most d plus one of the set's points. Constantin Caratheodory proved the theorem in 1911 for the case where the set is compact, and Ernst Steinitz extended it to arbitrary sets in 1914; two closely related theorems, due to Helly and Radon, can each be used to prove the other.

Facts

Statement

If a point x lies in the convex hull of a set P in d dimensional space, then x also lies in some d dimensional simplex whose vertices are drawn from P. 1

Proof Year

1911 1

Classification

Statement Form

Existence Theorem 1

Connections

Has Statement Form

Existence Theorem

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

In Branch

Discrete Geometry

Source Caratheodory's theorem (convex hull) (Wikipedia)

Sources

1. Caratheodory's theorem (convex hull) (Wikipedia)

Introductory section
if a point x lies in the convex hull Conv(P) of a set P in d dimensional space, then x lies in some d-dimensional simplex with vertices in P
Introductory section, historical attribution
The result is named for Constantin Caratheodory, who proved the theorem in 1911 for the case when P is compact.
In Branch: Discrete Geometry, Lead sentence

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Caratheodory's Theorem (Convex Hull)

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