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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
MathSciNet review: 962813
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Abstract: In a normed linear space $ X$, consider a nonempty closed set $ K$ which has the property that for some $ r > 0$ there exists a set of points $ {x_0} \in X\backslash K,d({x_0}K) > r$, which have closest points $ p({x_0}) \in K$ and where the set of points $ {x_0} - r(({x_0} - p({x_0}))/\vert\vert{x_0} - p({x_0})\vert\vert)$ is dense in $ X\backslash K$. If the norm has sufficiently strong differentiability properties, then the distance function $ d$ generated by $ K$ has similar differentiability properties and it follows that, in some spaces, $ K$ is convex.

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Keywords: Distance functions, metric projection, proximinal, Chebyshev sets, Gâteaux, Fréchet, uniformly Gâteaux, uniformly Fréchet differentiable
Article copyright: © Copyright 1988 American Mathematical Society

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