Czechoslovak Mathematical Journal, Vol. 68, No. 4, pp. 943-951, 2018


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

Tord Sjödin

Received January 30, 2017.   Published online May 3, 2018.

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 $P_i(x)\neq x_i$ and satisfies Hölder condition of order $1/(p-1)$ if $P_i(x)=x_i$.
Keywords:  normed space; uniform convexity; closed set; metric projection; $l^p$-space; Fréchet differential; Lipschitz condition
Classification MSC:  26E25, 46B20, 49J50


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Affiliations:   Tord Sjödin, Department of Mathematics, Umea University, Umea, 901 87, Sweden, e-mail: tord.sjodin@math.umu.se


 
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