Vertex of the Intersection of Nonempty Convex Sets

Question

If $x̂ ∈ R^n$ is a vertex of $C_1 ∩ C_2$, where $C_1∈R^n$ and $C_2∈R^n$ are nonempty convex sets, then $x̂$ is either a vertex of $C_1$ or a vertex of $C_2$.

Is this a claim valid? Is there any way we can prove/disprove this claim numerically?

Answer 1

This image should convince you that this is untrue for $n \ge 2$. The result is true for $n=1$ though.

Answer 2

For $n \ge 2$, the intersection of the closed balls $B_i$ ($i =1,2$) with radii equal to one and centers equal to $((-1)^i, 0, \dots, 0)$ is a single point: the origin. A point is a convex set with a vertex: itself. However the two balls are convex but don't have extreme points.