Carathéodory convex hull theorem

The Carathéodory convex hull theorem states that for any point of the convex hull of a subset of a Euclidean space, there exists a non-degenerate simplex containing it with vertices in this subset.

Another formulation: The convex hull of a finite-dimensional compact is a compact ^[1] .

Statement of the theorem

Let be $A\subset R^{m}$ ${\ displaystyle A \ subset R ^ {m}}$ Is a compact in m- dimensional Euclidean space . Then $\forall x\in co\ A$ ${\ displaystyle \ forall x \ in co \ A}$ (the convex hull of A) is a convex combination of at most m + 1 points in the set $A$ ${\ displaystyle A}$ ^[2] :

co\ A=\left\{x:x=\sum _{i=1}^{m+1}\lambda _{i}x_{i}(x),\quad x_{i}(x)\in A,\quad \lambda _{i}\geqslant 0,\quad \sum _{i=1}^{m+1}\lambda _{i}=1,\quad i=1,\ 2,\ \dots ,\ m+1\right\}

{\ displaystyle co \ A = \ left \ {x: x = \ sum _ {i = 1} ^ {m + 1} \ lambda _ {i} x_ {i} (x), \ quad x_ {i} ( x) \ in A, \ quad \ lambda _ {i} \ geqslant 0, \ quad \ sum _ {i = 1} ^ {m + 1} \ lambda _ {i} = 1, \ quad i = 1, \ 2, \ \ dots, \ m + 1 \ right \}}

Related Results

In the case when one of the coordinates of the point $x\in co\ A$ ${\ displaystyle x \ in co \ A}$ reaches an extreme value (for the set A ), this point can be represented as a convex combination of no more than m points A ^[2] .

Helly's theorem ^[2] is also connected with the Carathéodory convex hull theorem .

Notes

↑ § 1 Convex hulls. Lemma and Carathéodory's theorem
↑ ¹ ² ³ Yudin, 1974 , p. 22.

Literature

Yudin D. B. Mathematical control methods in conditions of incomplete information. - M .: "Soviet Radio", 1974. - 400 p.

[1] § 1 Convex hulls. Lemma and Carathéodory's theorem

[_22f78619babdb25f-2] ¹ ² ³ Yudin, 1974 , p. 22.

Carathéodory convex hull theorem

Statement of the theorem

Related Results

Notes

Literature

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