Infimum of norms of elements in a hyperplane

Question

In a Banach space X, given a norm one bounded linear functional $f$ and $c\in \mathbb{C}\backslash \{0\}$, define $H = \big\{ x\in X \,\vert\, f(x) = c\big\}$ and $\inf H$ = $\inf_{h\in H} \|h\|$.

1. Is there a necessary condition for $\inf H = \vert\, c\,\vert$?

Update: In $X$, we always have $\inf H = \vert\, c\,\vert$. Check this post.