Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 26 (2019), No. 2, 687--698Copyright Heldermann Verlag 2019 Convex Decompositions of Convex Open Sets with Polytopes or Finite Sets Removed Cornel Pintea Faculty of Mathematics and Computer Science, Babes-Bolyai University, 400084 M. Kogalniceanu 1, Cluj-Napoca, Romania cpintea@math.ubbcluj.ro [Abstract-pdf] We provide convex decompositions for the convex open sets with polytopes or finite sets removed, some of which are minimal in a certain sense. The valence of a function $f\colon O \rightarrow\mathbb{R}^n$, whose restrictions to all convex subsets of $O \subseteq \mathbb{R}^n$ are injective, cannot exceed the number of convex components of such decompositions. It is therefore worth to investigate the smallest number of convex subsets of $O$ needed to cover $O$. While the convex decompositions of the mentioned open complements are the main issue of this paper, a few remarks on this smallest number are provided by the end of the paper in the last section. Keywords: Convex decompositions, CIP-functions, valence of a function. MSC: 47H05; 47H99 [ Fulltext-pdf  (233  KB)] for subscribers only.