Convex sets and convex polytopes?

Question

Consider the set $\mathcal{X} \subset \mathbb{R}^d$ convex and compact. Which is the difference between the collection of compact convex subsets of $\mathcal{X}$ and the collection of convex polytopes of $\mathcal{X}$ (also in terms of cardinality)?

Answer 1

A convex polytope that is contained in $\mathcal{X}$ is described as the finite intersection of halfspaces; every such convex polytope is compact and convex. Thus the second class is a subset of the first class. However, the converse inclusion fails (consider a Euclidean ball intersected with the affine span of $\mathcal{X}$).

On the other hand, every compact convex subset of $\mathcal{X}$ can be approximated as a decreasing intersection of convex polytopes.

Altogether, the second class is a proper subset of the first class but both have the same cardinality.