Skip to main content
SpringerLink
Log in

On single-valuedness of convex set-valued maps

  • Published:
Set-Valued Analysis Aims and scope Submit manuscript

Abstract

It is proved that ifX andY are linear spaces andF :X →p(Y) is a set-valued map with convex graph such thatF(x) ≠ Ø for allx ∈X andF(x 0) is a singleton for somex 0, thenF is single-valued and affine. Applications to metric projections and to adjoints of set-valued maps are given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aubin, J. -P. and Ekeland, I.:Applied Nonlinear Analysis, Wiley, New York, 1984.

    Google Scholar 

  2. Aubin, J. -P. and Frankowska, H.:Set-Valued Analysis, Birkhäuser-Verlag, Boston, 1990.

    Google Scholar 

  3. Berge, C.:Topological Spaces, Oliver and Boyd, London, 1963 (see also the second French edition of this book:Espaces topologiques. Fonctions multivoques, Dunod, Paris, 1966).

    Google Scholar 

  4. Borwein, J. M.: Adjoint process duality,Math. Oper. Res. 8 (1983), 403–434.

    Google Scholar 

  5. Crouzeix, J. -P.: Continuité des applications linéaires multivoques,PAIRO 7 (1973), R-1, 62–67.

    Google Scholar 

  6. Dunford, N. and Schwartz, J. T.:Linear Operators I, Interscience, New York, 1958.

    Google Scholar 

  7. Gautier, S.: Différentiabilité des multiapplications, Publications de l'Université de Pau, 1978.

  8. Gautier, S. Affine and eclipsing multifunctions,Numer. Funct. Anal. Optim. 11 (1990), 679–699.

    Google Scholar 

  9. Ricceri, B.: Verbal communication with F. Deutsch, May 1991.

  10. Rockafellar, R.T.:Convex Analysis, Princeton University Press, Princeton, NJ, 1970.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Pennsylvania State University, 16802, University Park, PA, USA

    Frank Deutsch

  2. Institute of Mathematics of the Romanian Academy, PO Box 1-764, RO-70700, Bucharest, Romania

    Ivan Singer

Authors
  1. Frank Deutsch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ivan Singer
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Supported by NSF Grant DMS-9100228.

The main result of this paper has been obtained while the second author was visiting the Pennsylvania State University in the framework of the exchange agreement between the Romanian Academy and the National Academy of Sciences of the U.S.A.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Deutsch, F., Singer, I. On single-valuedness of convex set-valued maps. Set-Valued Anal 1, 97–103 (1993). https://doi.org/10.1007/BF01039295

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01039295

Key words

Search

Navigation