Probably the most intuitive theorem relating convexity and dimension; others are Helly's theorem and Radon's theorem.
Theorem. Let X be a convex set in Rd, and let x be some point in the interior of X. Then x is a convex combination of some d+1 points in X.
Equivalently, we may say that any convex combination of points in Rd is a convex combination of at most d+1 of them.
Here's a proof of Caratheodory's theorem on convexity.