A note on extreme points of $C^\infty$-smooth balls in polyhedral spaces

Abstract:

Morris (1983) proved that every separable Banach space $X$ that contains an isomorphic copy of $c_0$ has an equivalent strictly convex norm such that all points of its unit sphere $S_X$ are unpreserved extreme, i.e., they are no longer extreme points of $B_{X^{**}}$. We use a result of Hájek (1995) to prove that any separable infinite-dimensional polyhedral Banach space has an equivalent $C^{\infty }$-smooth and strictly convex norm with the same property as in Morris’ result. We additionally show that no point on the sphere of a $C^2$-smooth equivalent norm on a polyhedral infinite-dimensional space can be strongly extreme, i.e., there is no point $x$ on the sphere for which a sequence $(h_n)$ in $X$ with $\|h_n\|\not \to 0$ exists such that $\|x\pm h_n\|\to 1$.