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
StatementIf 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 Classification
Statement Form Caratheodory's Theorem (Convex Geometry)
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In Branch
Source Caratheodory's theorem (convex hull) (Wikipedia)
Principal Result Of
Source Caratheodory's theorem (convex hull) (Wikipedia)
Proved By
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.
Principal Result Of: Constantin Carathéodory, Lead sentence of the article on the result
The result is named for Constantin Carathéodory, who proved the theorem in 1911 for the case when P {\displaystyle P} is compact.
- In Branch: Discrete Geometry, Lead sentence
Proved By: Constantin Carathéodory, Lead
The result is named for Constantin Carathéodory, who proved the theorem in 1911
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