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 ∏_(K, P) from R~n (endowed with a polyhedral norm ‖·‖_P) to a polyhedral subset K of R~n. As a consequence, the two types of selections are piecewise affine selections for ∏_(K, P). Moreover, if ∏_(K, ∞) and ∏_(K, 1) are the metric projection onto K in R~n endowed with the l_∞-norm and the l_1-norm, respectively, we prove that ∏_(K, 1) has a piecewise affine and quasi-linear extremal point selection when K is a subspace, and that the strict best approximation sba_K(x) of x in K is a piecewise affine selection for ∏_(K, ∞).
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