Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Artstein, Z., Burns, J.: Integration of compact set-valued function. Pacific J. Math. 58, 297–307 (1975)
Artstein, Z.: On the calculus of closed set-valued functions. Indiana Univ. Math. J. 24(5), 433–441 (1975)
Aubin, J.-P., Frankowska, H.: Set-valued analysis. Birkhaüser Boston Inc., Boston (1990)
Aumann, R.J.: Integrals of set-valued functions. J. of Math. Anal. Appl. 12, 1–12 (1965)
Baier, R., Farkhi, E.: Regularity and integration of set-valued maps represented by generalized Steiner points. Set-Valued Anal. 15(2), 185–207 (2007)
Chistyakov, V.V.: Selections of bounded variation. J. Appl. Anal. 10, 1–82 (2004)
Dyn, N., Farkhi, E.: Set-valued approximations with Minkowski averages – convergence and convexification rates. Numer. Funct. Anal. Optim. 25, 363–377 (2004)
Dyn, N., Farkhi, E., Mokhov, A.: Approximations of set-valued functions by metric linear operators. Constr. Approx. 25, 193–209 (2007)
Dyn, N., Farkhi, E., Mokhov, A.: Approximations of set-valued functions (adaptation of classical approximation operators). Imperial College Press, London (2014)
Kels, S., Dyn, N.: Reconstruction of 3D objects from 2D cross-sections with the 4-point subdivision scheme adapted to sets. Comput. Graph. 35, 741–746 (2011)
Kolmogorov, A., Fomin, S.: Introductory real analysis. Dover Publication, New York (1975)
Mokhov, A.: Approximation and representation of set-valued functions with compact images. Tel-Aviv University, PhD Thesis (2011)
Natanson, I.P.: Theory of Functions of a Real Variable, vol. II. Frederick Ungar, New York (1960)
Rockafellar, R.T., Wets, R.: Variational analysis. Springer, Berlin (1998)
Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Cambridge University Press, Cambridge (1993)
Vitale, R.A.: Approximations of convex set-valued functions. J. Approximation Theory 26, 301–316 (1979)
Acknowledgments
This work is partially supported by the Hermann Minkowski Center for Geometry at Tel-Aviv University.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dyn, N., Farkhi, E. & Mokhov, A. The Metric Integral of Set-Valued Functions. Set-Valued Var. Anal 26, 867–885 (2018). https://doi.org/10.1007/s11228-017-0403-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11228-017-0403-1
Keywords
- Compact sets
- Set-valued functions
- Metric selections
- Metric linear combinations
- Aumann integral
- Kuratowski upper limit
- Metric integral