On almost everywhere differentiability of the metric projection on closed sets in $l^p\left({ℝ}^n\right)$, $2<p<\infty$

@article{Sjödin2018,
abstract = {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 $P_i(x)\ne x_i$ and satisfies Hölder condition of order $1/(p-1)$ if $P_i(x)=x_i$.},
author = {Sjödin, Tord},
journal = {Czechoslovak Mathematical Journal},
keywords = {normed space; uniform convexity; closed set; metric projection; $l^p$-space; Fréchet differential; Lipschitz condition},
language = {eng},
number = {4},
pages = {943-951},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On almost everywhere differentiability of the metric projection on closed sets in $l^p(\mathbb \{R\}^n)$, $2<p<\infty $},
url = {http://eudml.org/doc/294577},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Sjödin, Tord
TI - On almost everywhere differentiability of the metric projection on closed sets in $l^p(\mathbb {R}^n)$, $2<p<\infty $
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 4
SP - 943
EP - 951
AB - 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 $P_i(x)\ne x_i$ and satisfies Hölder condition of order $1/(p-1)$ if $P_i(x)=x_i$.
LA - eng
KW - normed space; uniform convexity; closed set; metric projection; $l^p$-space; Fréchet differential; Lipschitz condition
UR - http://eudml.org/doc/294577
ER -