Abstract

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 ⁣:O→Rnf\colon O \rightarrow\mathbb{R}^n, whose restrictions to all convex subsets of O⊆RnO \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 OO needed to cover OO. 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.

Author information

Contact details are reproduced from the original publication and may be historical.

Cornel Pintea

Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084 M. Kogălniceanu 1, Cluj-Napoca, Romania

cpintea@math.ubbcluj.ro

Keywords

  • Convex decompositions
  • CIP-functions
  • valence of a function

Mathematics Subject Classification

47H05; 47H99

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Suggested citation

C. Pintea. “Convex Decompositions of Convex Open Sets with Polytopes or Finite Sets Removed.” Journal of Convex Analysis 26 (2019), No. 2, 687–698. https://doi.org/10.68381/jca26036

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.