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Journal of Convex Analysis 07 (2000), No. 1, 073--094
Copyright Heldermann Verlag 2000

Piecewise Affine Selections for Piecewise Polyhedral Multifunctions and Metric Projections

M. Finzel
Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstr. 1 1/2, 91054 Erlangen, Germany

W. Li
Dept. of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529-0077, U.S.A.

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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