minimal distance betwen a point and and the halfspace containing a convex set

Question

Let $L^2(I)$ be the usual $L_2$ space with $L_2$ norm and $S$ a convex and compact subset of $L^2(I)$. Suppose $g^*\notin S$ and

$$\min_{f\in S} \|f-g^*\|$$

has the unique solution $f^*\in S$.

Suppose a supporting hyperplane $H\subset L_2(I)$ supports $S$ at $f^*$ and $S\subset H^{-}$.

Would it be true that

$$\min_{f\in H^{-}\cup H} \|f-g^*\|$$

also has a solution $f^*$?

Here compact may refer to weak compact.