Theorem

Caratheodory's Theorem (Convex Geometry)

Geometry

Caratheodory's Theorem states that if a point lies in the convex hull of a set of points in d-dimensional space, then that same point already lies in the convex hull of some subset of at most d plus one of the original points. Named for Constantin Caratheodory, it bounds how many points are ever needed to express any point of a convex hull as a combination of the original set, and is a foundational tool of convex geometry and linear programming. Other results are also named for Caratheodory in other branches of mathematics, including a measure-theory extension theorem and a complex-analysis boundary theorem, neither of which is minted on this atlas yet.

Facts

Statement

If a point x lies in the convex hull of a set P in d-dimensional space, then x lies in some d-dimensional simplex with vertices in 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)

Lead section
It states that if a point x lies in the convex hull Conv(P) of a set P
Naming and history paragraph
The result is named for Constantin Carathéodory, 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 Geometry)

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