is convex hull of intersection equal to intersection of convex hull

Question

is $convexhull(S_1\cap S_2)=convexhull(S_1)\cap convexhull(S_2)$

where $S_1$ and $S_2$ are finite sets.

Answer 1

The answer is no. For example the following two subsets of the plane

Answer 2

No, for example consider $S_1=\{1,3\}$, $S_2=\{2\}$. The intersection is empty, and the intersection of convex hulls is $S_2$.