Differentiability of distance functions and a proximinal property inducing convexity

Author:
J. R. Giles

Journal:
Proc. Amer. Math. Soc. **104** (1988), 458-464

MSC:
Primary 41A65; Secondary 46B20

DOI:
https://doi.org/10.1090/S0002-9939-1988-0962813-1

MathSciNet review:
962813

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Abstract: In a normed linear space , consider a nonempty closed set which has the property that for some there exists a set of points , which have closest points and where the set of points is dense in . If the norm has sufficiently strong differentiability properties, then the distance function generated by has similar differentiability properties and it follows that, in some spaces, is convex.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0962813-1

Keywords:
Distance functions,
metric projection,
proximinal,
Chebyshev sets,
Gâteaux,
Fréchet,
uniformly Gâteaux,
uniformly Fréchet differentiable

Article copyright:
© Copyright 1988
American Mathematical Society