Let $C = \{ e _{1}, \cdots e _{n} \}$, where each $e _{i}$ are unit vectors  in $\ell ^{p, \infty}$, and $1 < p < \infty$. I want prove that $\overline{conv}(C)$ is diametral. My doubt is: $\Vert x - y \Vert _{p, \infty} \leqslant 1$ for all $x, y \in \overline{conv}(C)$, where 
\begin{align*}
\Vert x \Vert _{p, \infty} = \max \{ \Vert x  ^{+}\Vert _{p}, \Vert x ^{-} \Vert _{p} \}
\end{align*}
and
\begin{align*}
(x ^{+}) ^{i} = \max \{ x _{i}, 0 \} = \frac{\vert x _{i}\vert + x _{i}}{2} \quad \mbox{and} \quad (x ^{-}) ^{i} = \max \{ -x _{i}, 0 \} = \frac{\vert x _{i}\vert - x _{i}}{2}
\end{align*}
for all $x _{i} \in \ell ^{p, \infty}$?.