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Nearest points to closed sets and directional derivatives of distance functions

Published online by Cambridge University Press:  17 April 2009

Simon Fitzpatrick
Simon Fitzpatrick
Affiliation:
Department of Mathematics and Statistics, University of Auckland, Auckland, New Zealand
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Abstract

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We investigate the circumstances under which the distance function to a closed set in a Banach space having a one-sided directional derivative equal to 1 or −1 implies the existence of nearest points. In reflexive spaces we show that at a dense set of points outside a closed set the distance function has a directional derivative equal to 1.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 39 , Issue 2 , April 1989 , pp. 233 - 238
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Borwein, J.M. and Preiss, D., ‘A smooth variational principle with applications to subdifferentiability of convex functions’, Trans. Amer. Math. Soc. 303 (1987), 517–527.CrossRefGoogle Scholar
[2]Borwein, J.M. and Giles, J.R., ‘The proximal normal formula in Banach spaces’, Trans. Amer. Math. Soc. 302 (1987), 371–381.CrossRefGoogle Scholar
[3]Borwein, J.M., Fitzpatrick, S.P. and Giles, J.R., ‘The differentiability of real valued functions on normed linear spaces using generalised subgradients’, J. Math. Anal. Appl. 128 (1987), 512–534.CrossRefGoogle Scholar
[4]Fitzpatrick, S., ‘Metric projections and the differentiability of distance functions’, Bull. Austral. Math. Soc. 22 (1980), 291–312.Google Scholar
[5]Fitzpatrick, S., ‘Differentiation of real valued functions and continuity of metric projections’, Proc. Amer. Math. Soc. 91 (1984), 544–648.Google Scholar
[6]Giles, J.R., ‘A distance function property implying differentiability’, Bull. Austral. Math. Soc. (to appear).Google Scholar
[7]James, R.C., ‘Weak compactness and reflexivity’, Israel Math. J. 2 (1964), 101–119.Google Scholar
[8]Lau, Ka Sing, ‘Almost Chebyshev subsets in reflexive Banach spaces’, Indiana Univ. Math. J. 2 (1978), 791–795.Google Scholar
[9]Preiss, D., ‘Differentiability of Lipschitz functions on Banach spaces’, (submitted).Google Scholar
[10]Zajicek, L., ‘Differentiability of distance functions and points of multivaluedness of the metric projection in Banach spaces’, Czech. Math. J. 33 (1983), 292–308.Google Scholar