Carathéodory's theorem (convex hull)

Carathéodory's theorem is a theorem in convex geometry. It states that if a point x {\displaystyle x} lies in the convex hull C o n v ( P ) {\displaystyle \mathrm {Conv} (P)} of a set P ⊂ R d {\displaystyle P\subset \mathbb {R} ^{d}} , then x {\displaystyle x} lies in some d {\displaystyle d} -dimensional simplex with vertices in P {\displaystyle P} .