| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2018-12-07T06:17:47Z |
| Last updated:
|
2021-01-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147513 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
|
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| Reference:
|
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| . |
