Set-Valued and Variational Analysis

, Volume 26, Issue 4, pp 867–885 | Cite as

The Metric Integral of Set-Valued Functions

  • Nira Dyn
  • Elza FarkhiEmail author
  • Alona Mokhov


This paper introduces a new integral of univariate set-valued functions of bounded variation with compact images in \({\mathbb {R}}^d\). The new integral, termed the metric integral, is defined using metric linear combinations of sets and is shown to consist of integrals of all the metric selections of the integrated multifunction. The metric integral is a subset of the Aumann integral, but in contrast to the latter, it is not necessarily convex. For a special class of segment functions equality of the two integrals is shown. Properties of the metric selections and related properties of the metric integral are studied. Several indicative examples are presented.


Compact sets Set-valued functions Metric selections Metric linear combinations Aumann integral Kuratowski upper limit Metric integral 

Mathematics Subject Classification (2010)

26E25 28B20 28C20 54C60 54C65 


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This work is partially supported by the Hermann Minkowski Center for Geometry at Tel-Aviv University.


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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel-Aviv UniversityTel AvivIsrael
  2. 2.Institute of Mathematics and Informatics Bulgarian Academy of SciencesSofiaBulgaria
  3. 3.Unit of MathematicsAfeka, Tel-Aviv Academic College of EngineeringTel AvivIsrael

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