What is the set of $x$ achieving the maximum in the definition of the support function $h_B(u)=\sup_{x\in B}\langle u,x\rangle$?

Question

Suppose that $B$ is a convex body (compact closed with nonempty interior), and let $$h_B(u) = \sup_{x \in B} \langle u,x\rangle$$ be its support function. Is there a nice description of the set $E :=\{x : \langle u,x\rangle = h_B(u)\}$, that is, the set of $x$ that achieve the supremum?

EDIT: Assume that $B$ is the unit ball of a norm $\|\cdot\|$. Can we relate the equality set $E$ above to the subdifferential of of $f(x) := \|x\|$?

Answer 1

$\{x\in B : \langle u,x\rangle = h_B(u)\}\subset\partial B$

Answer 2

The set $E$ is the supporting hyperplane for the set $B$ with normal $u$. In other words, it's the hyperplane normal to $u$ that has a non-zero intersection with $B$. If $B$ is strictly convex, then $E$ is a singleton.