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Piecewise affine functions and polyhedral sets. (English) Zbl 0816.49011
Summary: We present a number of characterizations of piecewise affine and piecewise linear functions defined on finite-dimensional normed vector spaces. In particular, we prove that a real-valued function is piecewise affine [resp., piecewise linear] if both its epigraph and its hypograph are (nonconvex) polyhedral sets [resp., polyhedral cones]. Also, we show that the collection of all piecewise affine [resp., piecewise linear] functions coincides with the smallest vector lattice containing the vector space of affine [resp. linear] functions. Furthermore, we prove that a function is piecewise affine [resp. piecewise linear] if it can be represented as a difference of two convex [resp. sublinear] polyhedral functions.

MSC:
49J52 Nonsmooth analysis
49J40 Variational inequalities
58C06 Set-valued and function-space-valued mappings on manifolds
90C31 Sensitivity, stability, parametric optimization
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