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On non-smooth convex distance functions

✍ Scribed by Ngoc-Minh Lê

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
617 KB
Volume
63
Category
Article
ISSN
0020-0190

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✦ Synopsis

Under the Euclidean metric in 3-space, the bisectors of three points intersect in at most one connected componentnamely, a line. In contrast to this, we show that, under non-smooth convex distance functions, there is no general upper bound to the number of connected components of the intersection of the bisectors of three points in 3-space. Our result is important for the further study of abstract Voronoi diagrams in 3-space, and -as a byproduct -disproves a conjecture of Schaudt and Drysdale (1992). @ 1997 Elsevier Science B.V.

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Let f be a continuous convex function on a Banach space E. This paper shows that every proper convex function g on E with g F f is generically Frechet differentiable if and only if the image of the subdifferential map Ѩ f of f has the Radon᎐Nikodym property, and in this case it is equivalent to show