# Locally piecewise affine functions and their order structure

Locally piecewise affine functions and their order structure Piecewise affine functions on subsets of $$\mathbb R^m$$ 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 $$\ddot{\mathrm{a}}$$ 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 $$C(\mathbb R^m)$$ C ( R m ) , while piecewise affine functions are sequentially order dense in $$C(\mathbb R^m)$$ C ( R m ) . This paper is partially based on Adeeb (Locally piece-wise affine functions, 2014) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Locally piecewise affine functions and their order structure

, Volume 21 (1) – Apr 11, 2016
9 pages

/lp/springer_journal/locally-piecewise-affine-functions-and-their-order-structure-iVBBgM4XK0
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-016-0411-7
Publisher site
See Article on Publisher Site

### Abstract

Piecewise affine functions on subsets of $$\mathbb R^m$$ 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 $$\ddot{\mathrm{a}}$$ 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 $$C(\mathbb R^m)$$ C ( R m ) , while piecewise affine functions are sequentially order dense in $$C(\mathbb R^m)$$ C ( R m ) . This paper is partially based on Adeeb (Locally piece-wise affine functions, 2014)

### Journal

PositivitySpringer Journals

Published: Apr 11, 2016

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