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Journal of Convex Analysis, Vol. 7, No. 1, pp. 73-94 (2000)
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Piecewise Affine Selections for Piecewise Polyhedral Multifunctions and Metric Projections

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M. Finzel and W. Li

Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstrasse 1 1/2, 91054 Erlangen, Germany, finzel@mi.uni-erlangen.de and Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529-0077, USA, wuli@math.odu.edu

**Abstract:** Piecewise polyhedral multifunctions are the set-valued version of piecewise affine functions. We investigate selections of piecewise polyhedral multifunctions, in particular, the least norm selection and continuous extremal point selections.

A special class of piecewise polyhedral multifunctions is the collection of metric projections $\Pi_{K,P}$ from $\mathbb{R}^n$ (endowed with a polyhedral norm $\|\cdot\|_P$) to a polyhedral subset $K$ of $\mathbb{R}^n$. As a consequence, the two types of selections are piecewise affine selections for $\Pi_{K,P}$. Moreover, if $\Pi_{K,\infty}$ and $\Pi_{K,1}$ are the metric projection onto $K$ in $\mathbb{R}^n$ endowed with the $\ell_\infty$-norm and the $\ell_1$-norm, respectively, we prove that $\Pi_{K,1}$ has a piecewise affine and quasi-linear extremal point selection when $K$ is a subspace, and that the strict best approximation $\operatorname{\mathbf{sba}}_K(x)$ of $x$ in $K$ is a piecewise affine selection for $\Pi_{K,\infty}$.

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