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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
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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

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The statement of the theorem only uses the linear structure of the ambient space, so the Banach space structure does not affect its validity.

One way of proving the theorem is by applying Helly's theorem. The latter seems to be more readily amenable to generalisations because of obvious connections to topology.

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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

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