Convex and discrete geometry Convex sets: basic notions Theorem

Carathéodory's theorem

Builds on Convex hull Affine combination

Statement

Let $P$ be a subset of $\mathbb{R}^d$ , and let $x$ be a point of its convex hull $\operatorname{conv}(P)$ . Then there are at most $d + 1$ points $p_1, \dots, p_k$ of $P$ , with $k \leq d + 1$ , such that $x \in \operatorname{conv}\{p_1, \dots, p_k\}$ . Equivalently, $x$ is a convex combination of at most $d + 1$ points of $P$ .

Remarks

Thus $\operatorname{conv}(P)$ is the union of the simplices with vertices in $P$ , of dimension at most $d$ . Mathlib states this form as convexHull_eq_union. The bound $d + 1$ is the best possible in general. Let $P$ be the three vertices of a triangle in $\mathbb{R}^2$ . A point inside the triangle is not in the hull of two points of $P$ . The proof only uses the order and the field operations of $\mathbb{R}$ . So it works over each ordered field.

History

Carathéodory proved the theorem in 1911 for compact sets $P$ . Steinitz extended it to arbitrary sets in 1914.

Reading path

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Tag	`0201Q4P2B5`
URI	`m2://0201Q4P2B5`
Kind	statement, theorem
Writing	W1 stated
MSC	52A35
Status	This copy of the site has no build data. Run `m2 status 0201Q4P2B5`.
Uses	Convex hull, Affine combination
Same as	Wikidata Q2471737 Wikipedia Carathéodory's theorem (convex hull)

Carathéodory's theorem

Statement

Remarks

History

Reading path

Where this leads

Further reading

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