Geodesic
On almost everywhere differentiability of the metric projection on closed sets in $l^p(\mathbb R^n)$, $2$
Czechoslovak Mathematical Journal, Tome 68 (2018) no. 4, pp. 943-951 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $2$ 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 $P_i(x)=x_i$.
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 $2$ 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 $P_i(x)=x_i$.
DOI : 10.21136/CMJ.2018.0038-17
Classification : 26E25, 46B20, 49J50
Keywords: normed space; uniform convexity; closed set; metric projection; $l^p$-space; Fréchet differential; Lipschitz condition 
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     author = {Sj\"odin, Tord},
     title = {On almost everywhere differentiability of the metric projection on closed sets in $l^p(\mathbb R^n)$, $2<p<\infty $},
     journal = {Czechoslovak Mathematical Journal},
     pages = {943--951},
     year = {2018},
     volume = {68},
     number = {4},
     doi = {10.21136/CMJ.2018.0038-17},
     mrnumber = {3881888},
     zbl = {07031689},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2018.0038-17/}
}
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Sjödin, Tord. On almost everywhere differentiability of the metric projection on closed sets in $l^p(\mathbb R^n)$, $2

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