
Journal of Convex Analysis 26 (2019), No. 1, 105116 Copyright Heldermann Verlag 2019 Vertices, Edges and Facets of the Unit Ball Francisco J. GarciaPacheco Dept. of Mathematics, University of Cadiz, 11519 Puerto Real, Spain garcia.pacheco@uca.es It has been recently proved that every real Banach space can be endowed with an equivalent norm in such a way that the new unit sphere contains a convex subset with nonempty interior relative to the unit sphere. In fact, under good conditions like separability or being weakly compactly generated, this renorming can be accomplished to have a dense amount of convex sets in the unit sphere with nonempty relative interior. Therefore, not all equivalent norms on a Banach space show some degree of strict convexity. In the opposite direction, for a long time it was unknown whether there exists a nonstrictly convex real Banach space of dimension strictly greater than 2 with a dense amount of extreme points in the unit sphere. This question has been recently solved in three dimensions. The idea behind this solution is to construct a 3dimensional unit ball whose boundary is made of extreme points except for two nontrivial segments (which are opposite to each other). This unit ball is a deformation of an ellipsoid. In this manuscript we follow this line of research and prove that every Banach space with dimension strictly greater than 2 admitting a strictly convex equivalent renorming admits a nonstrictly convex equivalent norm whose unit ball verifies that all of its proper faces are segments. Keywords: Vertices, edges, facets, faces, infinite dimensional Banach space, strictly convex, renorming. MSC: 46B20 [ Fulltextpdf (113 KB)] for subscribers only. 