Relative interior of a normal cone at a face of a convex polytope?

Question

Suppose $A$ is a nonempty convex polytope in $\mathbb{R}^n$. Suppose $F$ is a face of $A$.

Consider the normal cone of $A$ at $F$:
$C_A(F)=\{v\in\mathbb{R}^n:v\cdot x\geq v\cdot y\ \forall\ x\in F,y\in A\}$.

Is it true that $ri(C_A(F))=\{v\in\mathbb{R}^n:F=\arg\max_{x\in A}v\cdot x\}$? where $ri$ denotes the relative interior.

Answer 1

Yes, this is true, a direct proof can be found in Lemma 1 part (ii) of Piermont 2022.