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.