Locally piecewise affine functions and their order structure

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)