Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Francisco J. García-Pacheco

Dept. of Mathematics, University of Cadiz, 11519 Puerto Real, Spain

garcia.pacheco@uca.es

Keywords

  • Vertices
  • edges
  • facets
  • faces
  • infinite dimensional Banach space
  • strictly convex
  • renorming

Mathematics Subject Classification

46B20

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Suggested citation

F. J. García-Pacheco. “Vertices, Edges and Facets of the Unit Ball.” Journal of Convex Analysis 26 (2019), No. 1, 105–116. https://doi.org/10.68381/jca26007

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press.