Is the vector of a nonempty set also a vector of the intersection of nonempty sets?

Question

If $C_1 ⊆ R^n$ and $C_2 ⊆ R^n$ are two nonempty convex sets such that $C_1 ∩ C_2 ≠ ∅$, can we prove the following result? :

If $x̂ ∈ R^n$ is a vertex of $C_1$ such that $x̂ ∈ C_1 ∩ C_2$, then $x̂$ is also a vertex of $C_1 ∩ C_2$.

Answer 1

Simply check the definition:

If $x\in C_1\cap C_2$ is a vertex of $C_1$ then it does not lie in any open line segment connecting two points of $C_1$. Then it certainly does not lie in any open line segment connecting two points of $C_1\cap C_2$. Hence it is a vertex of $C_1\cap C_2$.