Arkiv för Matematik

  • Ark. Mat.
  • Volume 49, Number 2 (2011), 325-333.

On minimal exposed faces

Francisco Javier García-Pacheco

Full-text: Open access


In this paper we consider the problem of the non-empty intersection of exposed faces in a Banach space. We find a sufficient condition to assure that the non-empty intersection of exposed faces is an exposed face. This condition involves the concept of inner point. Finally, we also prove that every minimal face of the unit ball must be an extreme point and show that this is not the case at all for minimal exposed faces since we prove that every Banach space with dimension greater than or equal to 2 can be equivalently renormed to have a non-singleton, minimal exposed face.


The author wants to thank the referee for his valuable comments and suggestions.

Article information

Ark. Mat., Volume 49, Number 2 (2011), 325-333.

Received: 7 September 2009
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2010 © Institut Mittag-Leffler


García-Pacheco, Francisco Javier. On minimal exposed faces. Ark. Mat. 49 (2011), no. 2, 325--333. doi:10.1007/s11512-010-0123-3.

Export citation


  • Aizpuru, A. and García-Pacheco, F. J., A short note about exposed points in real Banach spaces, Acta Math. Sci. Ser. B Engl. Ed. 28 (2008), 797–800.
  • Kreĭn, M. and Milman, D., On extreme points of regular convex sets, Studia Math. 9 (1940), 133–138.
  • Phelps, R. R., Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math. 1364, Springer, Berlin–Heidelberg, 1989.