Abstract

We study various properties of Lipschitz continuous linear selectors on the family of all convex, nonempty and compact subsets of Rn\mathbb{R}^n. In particular, it is shown that if s is such a selector then the Lipschitz constant of s can be estimated from below by the norm of s(Bn)s(B^n), where BnB^n is the unit ball. A notion of a parametric representation of convex bodies is introduced and illustrated with examples.

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Krzysztof Przesławski

Instytut Matematyki, Politechnika Zielonogórska, ul. Podgórna 50, 65-246 Zielona Góra, Poland

References

  1. Z. Artstein: Extensions of Lipschitz selections and an application to differential inclusions, Nonlinear Anal. 16 (1991) 701-704.https://doi.org/10.1016/0362-546X(91)90176-2Google Scholar
  2. J. P. Aubin, A. Cellina: Differential Inclusions, Springer, Berlin, 1984.https://doi.org/10.1007/978-3-642-69512-4Google Scholar
  3. J. P. Aubin, H. Frankowska: Set-Valued Analysis, Birkhäuser, Boston, 1990.Google Scholar
  4. F. Deutsch, Wu Li, Sung-Ho Park: Characterizations of continuous and Lipschitz continuous selections for metric projections in normed linear spaces, J. Approx. Theory 58 (1989) 297-314.https://doi.org/10.1016/0021-9045(89)90031-2Google Scholar
  5. G. Dommisch: On the existence of Lipschitz-continuous and differentiable selections for multifunctions, Parametric optimization and related topics, (Proc. Int. Conf., Plauen, Thüringen, October 1985) Math. Res. 35, 60-73, Akademie-Verlag, Berlin, 1987.https://doi.org/10.1515/9783112720936-005Google Scholar
  6. E. Giné, M. G. Hahn: M-infinitely divisible random compact convex sets, In: Probability in Banach Spaces V, Lectures Notes in Math. 1153, Springer-Verlag, Berlin et al. (1985).https://doi.org/10.1007/BFb0074952Google Scholar
  7. M. Gromov: Convex sets and Kähler manifolds, In: Advances in Differential Geometry and Topology, ed. F. Tricerri, World Scientific, Singapore (1990) 1-38.https://doi.org/10.1142/9789814439381_0001Google Scholar
  8. K. Leichtweiss: Konvexe Mengen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1980.Google Scholar
  9. O. V. Levchuk: The Steiner derivative of locally Lipschitz functions, (Russian) Optimizatsiya 29 (46) (1982) 56-65.Google Scholar
  10. J. Lindenstrauss: On nonlinear projections in Banach spaces, Michigan Math. J. 11 (1964) 263-287.https://doi.org/10.1307/mmj/1028999141Google Scholar
  11. J. Mattioli: Minkowski operations and vector spaces, Set-Valued Analysis 3 (1995) 33-50.https://doi.org/10.1007/BF01033640Google Scholar
  12. W. Meyer: Characterization of the Steiner point, Pacific J. Math. 35 (1970) 717-725.https://doi.org/10.2140/pjm.1970.35.717Google Scholar
  13. Cz. Olech: Lexicographical order, range of integrals and "bang-bang" principle, In: Mathematical Theory of Control, Academic Press, New York (1967) 35-45.Google Scholar
  14. E. D. Positsel'skii: Characterization of Steiner points, (Russian) Mat. Zametki 14 (1973) 243-247.Google Scholar
  15. M. H. Protter, H. F. Weinberger: Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, N.J., 1967.Google Scholar
  16. K. Przestawski, D. Yost: Lipschitz retracts, selectors, and extensions, Michigan Math. J. 42 (1995) 555-571.https://doi.org/10.1307/mmj/1029005313Google Scholar
  17. R. T. Rockafellar: Convex Analysis, Princeton Univ. Press, Princeton, 1970.https://doi.org/10.1515/9781400873173Google Scholar
  18. J. Saint-Pierre: Point de Steiner et sections lipschitziennes, Sém. Anal. Convexe 15(7) (1985).Google Scholar
  19. R. Schneider: On Steiner points of convex bodies, Israel J. Math. 9 (1971) 241-248.https://doi.org/10.1007/BF02771589Google Scholar
  20. G. C. Shephard: The Steiner point of a convex polytope, Canad. J. Math. 19 (1967) 1294-1300.https://doi.org/10.4153/CJM-1966-128-4Google Scholar
  21. W. Spiegel: Zur Minkowski-Additivität bestimmter Eikörperabbildungen, J. Reine Angew. Math. 286/287 (1976) 164-168.https://doi.org/10.1515/crll.1976.286-287.164Google Scholar
  22. G. Stefani, P. Zecca: Multivalued differential equations on manifolds with application to control theory, Illinois Math. J. 24 (1980) 560-575.https://doi.org/10.1215/ijm/1256047473Google Scholar
  23. E. M. Stein, G. Weiss: Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, 1971.https://doi.org/10.1515/9781400883899Google Scholar
  24. R. A. Vitale: The Steiner point in infinite dimensions, Israel J. Math. 52 (1985) 245-250.https://doi.org/10.1007/BF02786520Google Scholar

Suggested citation

K. Przesławski. “Lipschitz Continuous Selectors. Part I: Linear Selectors.” Journal of Convex Analysis 5 (1998), No. 2, 249–267. https://doi.org/10.68381/jca05-18

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Published by Heldermann Verlag, 1998. Rights now held by Banach Press LLC.