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)
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Acknowledgments
We would like to thank Foivos Xanthos for valuable discussions.
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The authors were supported by NSERC grants.
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Adeeb, S., Troitsky, V.G. Locally piecewise affine functions and their order structure. Positivity 21, 213–221 (2017). https://doi.org/10.1007/s11117-016-0411-7
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DOI: https://doi.org/10.1007/s11117-016-0411-7