A generalization of carathéodory's theorem

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Abstract

The following theorem is proved. If the sets V1,…, Vn+1Rn and a ϵ⋓n+1i=1 conv Vi, then there exist elements vi ϵ Vi (i=1…,n+1) such that a ϵ conv{v1,…,vn+1}. This is generalization of Carathéodory's theorem. By applying this and similar results some open questions are answered.

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