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Metric projections and the differentiability of distance functions

Published online by Cambridge University Press:  17 April 2009

Simon Fitzpatrick
Simon Fitzpatrick
Affiliation:
Department of Mathematics, University of Washington, Seattle, Washington 98195, USA.
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Abstract

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Let M be a closed subset of a Banach space E such that the norms of both E and E* are Fréchet differentiable. It is shown that the distance function d(·, M) is Fréchet differentiable at a point x of E ∼ M if and only if the metric projection onto M exists and is continuous at X. If the norm of E is, moreover, uniformly Gateaux differentiable, then the metric projection is continuous at x provided the distance function is Gateaux differentiable with norm-one derivative. As a corollary, the set M is convex provided the distance function is differentiable at each point of E ∼ M. Examples are presented to show that some of our hypotheses are needed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 22 , Issue 2 , October 1980 , pp. 291 - 312
Copyright
Copyright © Australian Mathematical Society 1980

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