A new convexity property that implies a fixed point property for $L_{1}$

Lennard, Chris

Studia Mathematica, Tome 100 (1991), p. 95-108 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 1991-01-01

Zbl 0762.46007

EUDML-ID : urn:eudml:doc:215881

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