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

Abstract

Morris [Mo83] proved that every separable Banach space X that contains an isomorphic copy of c0 has an equivalent strictly convex norm such that all points of its unit sphere SX are unpreserved extreme, i.e., they are no longer extreme points of BX * * . We use a result of Hájek [Ha95] to prove that any separable infinite-dimensional polyhedral Banach space has an equivalent C ∞ -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 (hn) in X with ∥hn∥ ̸ → 0 exists such that ∥x ± hn∥ → 1.