Geodesic
A new convexity property that implies a fixed point property for $L_{1}$
Studia Mathematica, Tome 100 (1991) no. 2, pp. 95-108

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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.
DOI : 10.4064/sm-100-2-95-108
Keywords: uniform Kadec-Klee property, convergence in measure compact sets, convex sets, normal structure, Lebesgue function spaces, fixed point, nonexpansive mapping, Chebyshev centre

Chris Lennard 1

1 
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Chris Lennard. A new convexity property that implies a fixed point property for $L_{1}$. Studia Mathematica, Tome 100 (1991) no. 2, pp. 95-108. doi: 10.4064/sm-100-2-95-108

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