Geodesic
Piecewise Affine Selections for Piecewise Polyhedral Multifunctions and Metric Projections
Journal of convex analysis, Tome 7 (2000) no. 1, pp. 73-94

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Piecewise polyhedral multifunctions are the set-valued version of piecewise affine functions. We investigate selections of such 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 from the n-dimensional euclidean space endowed with a polyhedral norm to a polyhedral subset K. As a consequence, the two types of selections are piecewise affine selections. Moreover, if we consider the euclidean setting endowed with the 1-norm, then the metric projection has a piecewise affine and quasi-linear extremal point selection when K is a subspace; and if we take the maximum norm then the strict best approximation is a piecewise affine selection for the metric projection onto the polyhedral subset K. 
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     author = {M. Finzel and W. Li},
     title = {Piecewise {Affine} {Selections} for {Piecewise} {Polyhedral} {Multifunctions} and {Metric} {Projections}},
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M. Finzel; W. Li. Piecewise Affine Selections for Piecewise Polyhedral Multifunctions and Metric Projections. Journal of convex analysis, Tome 7 (2000) no. 1, pp. 73-94. http://geodesic.mathdoc.fr/item/JCA_2000_7_1_JCA_2000_7_1_a3/