real analysis – Does the linear span of norming extreme points contain all extreme points?


Let $X$ be a Banach space, $X^*$ be its continuous dual space, and $B(X^*)$ be its closed unit ball, i.e., $B(X^*) = {x^*in X^*: |x^*|leq 1}$.

We say that $x^*$ is an extreme point of $B(X^*)$ if $x^*in B(X^*)$ and for any $x^* = frac{1}{2}(x_1^* + x_2^*)$ for some $x_1^*,x_2^*in B(X)$, we have $x^* =x_1^*=x_2^*$.

We say that $x^*$ is a norming extreme point of $B(X)$ if $x^*$ is an extreme point of $B(X)$ that satisfies $x^*(x) = |x|$ for some $xin Xsetminus {0}$.

Question: If $x^*$ is an extreme point of $B(X)$, then is it true that
$$x^* = a_1x_1^* + cdots +a_n x_n^*$$
for some $ngeq 1$, $a_1,…,a_nin mathbb{R}$ and $x_1^*,…,x_n^*$ are norming extreme points of $B(X^*)$?