A new convexity property that implies a fixed point property for L 1

Chris Lennard

Studia Mathematica (1991)

  • Volume: 100, Issue: 2, page 95-108
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Lennard, Chris. "A new convexity property that implies a fixed point property for $L_{1}$." Studia Mathematica 100.2 (1991): 95-108. <http://eudml.org/doc/215881>.

@article{Lennard1991,
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.},
author = {Lennard, Chris},
journal = {Studia Mathematica},
keywords = {uniform Kadec-Klee property; convergence in measure compact sets; convex sets; normal structure; Lebesgue function spaces; fixed point; nonexpansive mapping; Chebyshev centre; Kadec-Klee property; weak convergence; convergence of norms; fixed point property for nonexpansive mappings in convex sets},
language = {eng},
number = {2},
pages = {95-108},
title = {A new convexity property that implies a fixed point property for $L_\{1\}$},
url = {http://eudml.org/doc/215881},
volume = {100},
year = {1991},
}

TY - JOUR
AU - Lennard, Chris
TI - A new convexity property that implies a fixed point property for $L_{1}$
JO - Studia Mathematica
PY - 1991
VL - 100
IS - 2
SP - 95
EP - 108
AB - 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.
LA - eng
KW - uniform Kadec-Klee property; convergence in measure compact sets; convex sets; normal structure; Lebesgue function spaces; fixed point; nonexpansive mapping; Chebyshev centre; Kadec-Klee property; weak convergence; convergence of norms; fixed point property for nonexpansive mappings in convex sets
UR - http://eudml.org/doc/215881
ER -

References

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