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Journal of Convex Analysis 26 (2019), No. 2, 687--698
Copyright 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


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

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