Let $f_C$ be the convex envelope of $f$ on a non-empty convex set $C \subset \mathbb{R}^n$
I need to show that $\{ x^∗ \in C : f(x^∗) \leq f(x), \forall x \in C \} \subset \{x^∗ \in C : f_C(x^∗) \leq f_C(x), \forall x \in C \}.$
Right now, I can show that: $\{ x^∗ \in C : f(x^∗) \leq f(x), \forall x \in C \} \subseteq \{x^∗ \in C : f_C(x^∗) \leq f_C(x) \forall x \in C\}$
However, I need a counterexample to show that the equality in the subset relation does not hold.
For reference: The convex envelope of $f$ on $C$, denoted $f_C : C \rightarrow R$ is a convex function such that $f_C(x) \leq f(x) \forall x \in C$. The definition also requires that if $g$ is any other convex function on $C$ for which $g(x) \leq f(x), \forall x \in C$ then $f_C(x) \geq g(x), \forall x \in C$.
EDIT: Sorry for this, turns out that there was a typo and the $\subset$ should have actually been a $\subseteq$. After all, f could be a convex function in which case, it would be the same as it's convex hull.