Proyecciones Journal of Mathematics Vol. 31, N^o 2, pp. 91-101, June 2012. Universidad Católica del Norte Antofagasta - Chile

 

Asymptotically convex Banach spaces and The Index of Rotundity Problem

 

Francisco J. García-Pacheco

University of Cádiz, The European Union

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ABSTRACT

The Index of Rotundity Problem asks whether a Banach space which admits equivalent renormings with index of rotundity as small as desired also admits an equivalent rotund renorming. In this paper we continue the ongoing search for a negative answer to this question by making use of a new concept: asymptotically convex Banach spaces. Some applications to The Approximation Hyperplane Series Property are given.

Subjclass : [2000] Primary 46B20, 46B10.

Keywords : Rotund,renorming,Banach space.

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REFERENCES

[1] M.D.Acosta, R.M. Aron, D.Garcia, and M. Maestre, The Bishop-Phelps-Bollobás Theorem for operators, J. Funct. Anal. 254 11, pp. 2780—2799,(2008).

[2] M.D. Acosta, R.M. Aron, F.J. Garcia-Pacheco, Somethingrare is going on with the Approximation Hyperplane Series Property, Preprint.

[3] R. Deville, G. Godefroy and V. Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64, Longman Scientific, New York, (1993).

[4] J. Lindenstrauss, On operators which attain their norms, Israel J. Math. 1, pp. 139—148, (1963).

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Received : December 2011. Accepted : January 2012

Francisco J. García-Pacheco

Department of Mathematics

University of Cadiz

Puerto Real (ESP) 11510

The European Union

e-mail : garcia.pacheco@uca.es