I have the following conjecture:
Given a closed convex set $S \subseteq \mathbb{R}^n$ and one of its exposed face $F=\{x \in S \mid \pi x = \pi_0\}$, where $\pi x =\pi_0$ is the supporting hyperplane of $S$. For any $\delta>0$, the neighborhood $F_\delta$ of $F$ is defined as: $F_\delta = \{x \in S \mid \exists x' \in F \text{ s.t. } \|x'-x\|_2 < \delta\}$. Then there exists $\epsilon>0$, such that for any $\pi'$ with $\|\pi' - \pi\| < \epsilon, \sup_{x \in S} \langle \pi', x \rangle =\sup_{x \in F_\delta} \langle \pi', x \rangle$.
This conjecture is correct if $S$ is a polyhedron or $F$ is compact. For general (unbounded) set $S$ however, I really have no clue... In fact for $S$ being polyhedron, we have a stronger result: for any vector $\pi'$ close enough to $\pi$, the optimal solution of $\sup_{x \in S} \langle \pi', x \rangle$ with be $\textbf{contained}$ in $F$ (this is also known as "sticky face lemma").
I believe this conjecture is intuitively correct, and I think it should appear somewhere in the literature probably in some other forms. Any guidance or references will be much appreciated!
