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- Title
Locally piecewise affine functions and their order structure.
- Authors
Adeeb, S.; Troitsky, V.
- Abstract
Piecewise affine functions on subsets of $$\mathbb R^m$$ were studied in Aliprantis et al. (Macroecon Dyn 10(1):77-99, 2006), Aliprantis et al. (J Econometrics 136(2):431-456, 2007), Aliprantis and Tourky (Cones and duality, 2007), Ovchinnikov (Beitr $$\ddot{\mathrm{a}}$$ ge Algebra Geom 43:297-302, 2002). In this paper we study a more general concept of a locally piecewise affine function. We characterize locally piecewise affine functions in terms of components and regions. We prove that a positive function is locally piecewise affine iff it is the supremum of a locally finite sequence of piecewise affine functions. We prove that locally piecewise affine functions are uniformly dense in $$C(\mathbb R^m)$$ , while piecewise affine functions are sequentially order dense in $$C(\mathbb R^m)$$ . This paper is partially based on Adeeb (Locally piece-wise affine functions, 2014)
- Subjects
PIECEWISE affine systems; LATTICE theory; ORDER; MATHEMATICAL domains; AFFINE algebraic groups
- Publication
Positivity, 2017, Vol 21, Issue 1, p213
- ISSN
1385-1292
- Publication type
Article
- DOI
10.1007/s11117-016-0411-7