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Title: On almost everywhere differentiability of the metric projection on closed sets in $l^p(\mathbb R^n)$, $2<p<\infty $ (English)
Author: Sjödin, Tord
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 68
Issue: 4
Year: 2018
Pages: 943-951
Summary lang: English
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Category: math
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Summary: Let $F$ be a closed subset of $\mathbb R^n$ and let $P(x) $ denote the metric projection (closest point mapping) of $x\in \mathbb R^n$ onto $F$ in $l^p$-norm. A classical result of Asplund states that $P$ is (Fréchet) differentiable almost everywhere (a.e.) in $\mathbb R^n$ in the Euclidean case $p=2$. We consider the case $2<p<\infty $ and prove that the $i$th component $P_i(x)$ of $P(x)$ is differentiable a.e.\ if $P_i(x)\neq x_i$ and satisfies Hölder condition of order $1/(p-1)$ if $P_i(x)=x_i$. (English)
Keyword: normed space
Keyword: uniform convexity
Keyword: closed set
Keyword: metric projection
Keyword: $l^p$-space
Keyword: Fréchet differential
Keyword: Lipschitz condition
MSC: 26E25
MSC: 46B20
MSC: 49J50
idZBL: Zbl 07031689
idMR: MR3881888
DOI: 10.21136/CMJ.2018.0038-17
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Date available: 2018-12-07T06:17:47Z
Last updated: 2020-01-05
Stable URL: http://hdl.handle.net/10338.dmlcz/147513
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