[Math] Is the intersection of convex hulls a convex hull

Question

Given two finite sets of points, $X$ and $Y$, in $\mathbb R^d$ and assuming that $\text{conv}(X)\cap\text{conv}(Y)\neq\varnothing$.

I would guess that the intersection is a convex hull of some other finite set of points, $Z\in\mathbb R^d$ but is this actually true? How would I show it?

Answer 1

Let $Z$ be the set of extreme points of $\text{conv}(X) \cap \text{conv}(Y)$. Since the intersection of two convex sets is convex you have $$ \text{conv}(X) \cap \text{conv}(Y) = \text{conv}( \text{conv}(X) \cap \text{conv}(Y) ) = \text{conv}(Z).$$