Carathéodory's theorem (convex hull)

Carathéodory's theorem is a theorem in convex geometry. It states that if a point ${\displaystyle x}$ lies in the convex hull ${\displaystyle \mathrm {Conv} (P)}$ of a set ${\displaystyle P\subset \mathbb {R} ^{d}}$, then ${\displaystyle x}$ lies in some ${\displaystyle d}$-dimensional simplex with vertices in ${\displaystyle P}$. Equivalently, ${\displaystyle x}$ can be written as the convex combination of at most ${\displaystyle d+1}$ points in ${\displaystyle P}$. Additionally, ${\displaystyle x}$ can be written as the convex combination of at most ${\displaystyle d+1}$ extremal points in ${\displaystyle P}$, as non-extremal points can be removed from ${\displaystyle P}$ without changing the membership of ${\displaystyle x}$ in the convex hull.

An equivalent theorem for conical combinations states that if a point ${\displaystyle x}$ lies in the conical hull ${\displaystyle \mathrm {Cone} (P)}$ of a set ${\displaystyle P\subset \mathbb {R} ^{d}}$, then ${\displaystyle x}$ can be written as the conical combination of at most ${\displaystyle d}$ points in ${\displaystyle P}$.^: 257

Two other theorems of Helly and Radon are closely related to Carathéodory's theorem: the latter theorem can be used to prove the former theorems and vice versa.

The result is named for Constantin Carathéodory, who proved the theorem in 1911 for the case when ${\displaystyle P}$ is compact. In 1914 Ernst Steinitz expanded Carathéodory's theorem for arbitrary sets.