Unit balls of Polyhedral Banach spaces with many extreme points

Abstract

Let $E$ be a $(\mathrm{IV})$-polyhedral Banach space. We show that, for each $\epsilon>0$, $E$ admits an $\epsilon$-equivalent $\mathrm{(V)}$-polyhedral norm such that the corresponding closed unit ball is the closed convex hull of its extreme points. In particular, we obtain that every separable isomorphically polyhedral Banach space, for each $\epsilon>0$, admits an $\epsilon$-equivalent $(\mathrm{V})$-polyhedral norm such that the corresponding closed unit ball is the closed convex hull of its extreme points.