## Cornel Pintea, Adrian Tofan, Convex decompositions and the valence of some functions

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DOI: 10.23952/jnva.4.2020.2.05
Volume 4, Issue 2, 1 August 2020, Pages 225-239

Abstract. The aim of this paper is twofold. On one hand, we provide examples of $\mathbb{R}^n$-valued functions on some open subsets $D$ of $\mathbb{R}^n$ whose restrictions to the convex subsets of $D$ are all injective. Such applications are shortly called CIP functions. On the other hand, we provide alternative descriptions of the maximal convex subsets of the convex open sets with compact convex subsets removed. The maximal convex subsets of $\mathbb{R}^n$ with convex sets removed were characterized before by Martínez-Legaz and Singer [Compatible Preorders and Linear Operators on $\mathbb{R}^n$, Linear Algebra Appl. 153 (1991), 53-66] as being the convex subsets of $\mathbb{R}^n$, shortly called hemispaces, whose complements are convex too. The two topics merge together as the smallest number $k$ of convex subsets, of the considered open set, needed to cover it, is an upper bound for the valence of every CIP function on that open set.