Loterre: Mathematics: Carathéodory's theorem

geometry > convex analysis > convex set > Carathéodory's theorem

geometry > discrete geometry > Carathéodory's theorem

Carathéodory's theorem

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 ${\\displaystyle x}$ can be written as the convex combination of at most ${\\displaystyle d+1}$ points in ${\\displaystyle P}$ . More sharply, ${\\displaystyle x}$ 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.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_theorem_(convex_hull))

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