# Extensions of Carathéodory's theorem

We know about the Carathéodory's theorem which is on the convex bodies of $\mathbb{R}^d$. My question is, how far we can extend it? Is it true for say, any convex object of Banach space, or for convex objects of any real manifold?

I believe the answers are negative. However I want to know whether there is a similar result like Carathéodory (i.e. an upper bound with respect to dimension of the space/manifold). I am not sure whether it should be asked here. I have already asked it in math.stackexchange without getting any reply. The only comment was regarding the Choquet Theory, but I am yet to get anything related to my question. I am new to this subject, and possibly overlooked the required section. Advanced thanks for any help/suggestion/reference which can be (relatively) easily understood. Feel free to ask (and also edit) if you want more clarification.

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Perhaps we need a definition of "convex" in a manifold. –  Gerald Edgar Apr 14 '13 at 16:44
Roughly speaking, Caratheodory's theorem states that the convex hull of a set $X$ is the union of the simplices whose vertices lie on $X$. To obtain an extension in this sense you need to generalize the notion of simplex. This is what Choquet did: encyclopediaofmath.org/index.php/Choquet_simplex That explains the relation of Choquet theory to your question. –  alvarezpaiva Apr 14 '13 at 17:31
That is, in a $d$-dimensional affine subspace, everything in the theorem for $\mathbb R^d$ still works, for the reasons cited. –  Gerald Edgar Apr 14 '13 at 16:43