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A distance function property implying differentiability

Published online by Cambridge University Press:  17 April 2009

J.R. Giles
J.R. Giles
Affiliation:
Department of Mathematics, The University of Newcastle, New South Wales 2308, Australia
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Abstract

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In a real normed linear space X, properties of a non-empty closed set K are closely related to those of the distance function d which it generates. If X has a uniformly Gâteaux (uniformly Fréchet) differentiable norm, then d is Gâteaux (Fréchet) differentiable at x ∈ X/K if there exists an such that

and is Géteaux (Fréchet) differentiable on X / K if there exists a set P+(K) dense in X/K where such a limit is approached uniformly for all x ∈ P+(K). When X is complete this last property implies that K is convex.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 39 , Issue 1 , February 1989 , pp. 59 - 70
Copyright
Copyright © Australian Mathematical Society 1989

References

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