# Article

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Keywords:
normed space; uniform convexity; closed set; metric projection; $l^p$-space; Fréchet differential; Lipschitz condition
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 $P_i(x)=x_i$.
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