Vertices, Edges and Facets of the Unit Ball

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 non-empty 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 non-empty 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 non-strictly 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 3-dimensional unit ball whose boundary is made of extreme points except for two non-trivial 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 non-strictly convex equivalent norm whose unit ball verifies that all of its proper faces are segments.