# A new convexity property that implies a fixed point property for ${L}_{1}$

Studia Mathematica (1991)

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

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## 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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5. [D-S] D. van Dulst and B. Sims, Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK), in: Banach Space Theory and its Applications, Proc. Bucharest 1981, Lecture Notes in Math. 991, Springer, 1983, 35-43.
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11. [Ki₁] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. Zbl0141.32402
12. [Ki₂] W. A. Kirk, An abstract fixed point theorem for nonexpansive mappings, Proc. Amer. Math. Soc. 82 (1981), 640-642. Zbl0471.54027
13. [K-F] A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Dover Publ., 1975.
14. [L-T] E. Lami Dozo and Ph. Turpin, Nonexpansive maps in generalized Orlicz spaces, Studia Math. 86 (1987), 155-188. Zbl0649.47044
15. [L-M] A. T. Lau and P. F. Mah, Normal structure in dual Banach spaces associated with a locally compact group, Trans. Amer. Math. Soc. 310 (1988), 341-353. Zbl0706.43003
16. [L₁] C. J. Lennard, Operators and geometry of Banach spaces, Ph.D. dissertation, 1988.
17. [L₂] C. J. Lennard, C₁ is uniformly Kadec-Klee, Proc. Amer. Math. Soc. 109 (1990), 71-77.
18. [Pa] J. R. Partington, On nearly uniformly convex Banach spaces, Math. Proc. Cambridge Philos. Soc. 93 (1983), 127-129. Zbl0507.46011
19. [Pe] J. P. Penot, Fixed point theorems without convexity, in: Analyse Non Convexe (Pau 1977), Bull. Soc. France Mem. 60 (1979), 129-152.

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