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Title

A new convexity property that implies a fixed point property for L1

Authors 1

Affiliations

  1. Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, U.S.A.

Abstract

In this paper we prove a new convexity property for L₁ that resembles uniform convexity. We then develop a general theory that leads from the convexity property through normal structure to a fixed point property, via a theorem of Kirk. Applying this theory to L₁, we get the following type of normal structure: any convex subset of L₁ of positive diameter that is compact for the topology of convergence locally in measure, must have a radius that is smaller than its diameter. Indeed, a stronger result holds. The Chebyshev centre of any norm bounded, convergence locally in measure compact subset of L₁ must be norm compact. Immediately from normal structure, we get a new proof of a fixed point theorem for L₁ due to Lami Dozo and Turpin.

Keywords

ENG
uniform Kadec-Klee property, convergence in measure compact sets, convex sets, normal structure, Lebesgue function spaces, fixed point, nonexpansive mapping, Chebyshev centre

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Studia Mathematica
1991/100/2
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Pages:
95-108
Main language of publication
English
Received
1990-07-31
Accepted
1990-11-27
Published
1991
Exact and natural sciences