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The Australian Journal of Mathematical Analysis and Applications
ISSN 1449-5910 |
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Paper Title:
Differentiability of
Distance Functions in p-Normed Spaces
Author(s):
M. S. Moslehian, A. Niknam, S. Shadkam Torbati
Department of Pure Mathematics,
Centre of Excellence in Analysis on Algebraic Structures (CEAAS),,
Ferdowsi University of Mashhad,
P. O. Box
1159, Mashhad,
Iran
moslehian@ferdowsi.um.ac.ir
niknam@math.um.ac.ir
shadkam.s@wali.um.ac.ir
Abstract:
The farthest point mapping in a p-normed space X is studied in virtue of the Gateaux derivative and the Frechet derivative. Let M be a closed bounded subset of X having the uniformly p-Gateaux differentiable norm. Under certain conditions, it is shown that every maximizing sequence is convergent, moreover, if M is a uniquely remotal set then the farthest point mapping is continuous and so M is singleton. In addition, a Hahn--Banach type theorem in $p$-normed spaces is proved.
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