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arXiv · 1603.04897

Locally piecewise affine functions and their order structure

Samer Adeeb·Vladimir G. Troitsky
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Abstract

Piecewise affine functions on subsets of $\mathbb R^m$ were studied in \cite{Ovchinnikov:02,Aliprantis:06a,Aliprantis:07a,Aliprantis:07}. 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 \cite{Adeeb:14}.

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Samer Adeeb, Vladimir G. Troitsky. 2016-03-15. Locally piecewise affine functions and their order structure. https://arxiv.org/abs/1603.04897

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