Metric projections and the differentiability of distance functions
1980 ◽
Vol 22
(2)
◽
pp. 291-312
◽
Keyword(s):
Let M be a closed subset of a Banach space E such that the norms of both E and E* are Fréchet differentiable. It is shown that the distance function d(·, M) is Fréchet differentiable at a point x of E ∼ M if and only if the metric projection onto M exists and is continuous at X. If the norm of E is, moreover, uniformly Gateaux differentiable, then the metric projection is continuous at x provided the distance function is Gateaux differentiable with norm-one derivative. As a corollary, the set M is convex provided the distance function is differentiable at each point of E ∼ M. Examples are presented to show that some of our hypotheses are needed.
1998 ◽
Vol 3
(1-2)
◽
pp. 85-103
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1983 ◽
Vol 33
(2)
◽
pp. 292-308
◽
2006 ◽
Vol 02
(03)
◽
pp. 431-453
2021 ◽
Vol 5
◽
pp. 82-92
1992 ◽
Vol 5
(4)
◽
pp. 363-373
◽
Keyword(s):
1981 ◽
Vol 89
(1-2)
◽
pp. 75-86
◽
Keyword(s):
1999 ◽
Vol 22
(1)
◽
pp. 217-220
1985 ◽
Vol 31
(3)
◽
pp. 421-432
◽