Embedding theorems for classes of convex sets

Abstract

Rådström's embedding theorem states that the nonempty compact convex subsets of a normed vector space can be identified with points of another normed vector space such that the embedding map is additive, positively homogeneous, and isometric. In the present paper, extensions of Rådström's embedding theorem are proven which provide additional information on the embedding space. These results include those of Hörmander who proved a similar embedding theorem for the nonempty closed bounded convex subsets of a Hausdorff locally convex vector space. In contrast to Hörmander's approach via support functionals, all embedding theorems of the present paper are proven by a refinement of Rådström's original method which is constructive and does not rely on Zorn's lemma. This paper also includes a brief discussion of some actual or potential applications of embedding theorems for classes of convex sets in probability theory, mathematical economics, interval mathematics, and related areas.

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References

  1. 1.

    Abid, M.: ‘Un théorème ergodique pour des processus sous-additifs et sur-stationnaires’, C.R. Acad, Sci. Paris Série A 287 (1978), 149–152.

    Google Scholar 

  2. 2.

    Aliprantis, C. D., and Brown, D. J.: ‘Equilibria in Markets with a Riesz Space of Commodities’, J. Math. Econ. 11 (1983), 189–207.

    Google Scholar 

  3. 3.

    Aliprantis, C. D., and Burkinshaw, O.: Locally Solid Riesz Spaces, Academic Press, New York, San Francisco, London, 1978.

    Google Scholar 

  4. 4.

    Arrow, K. J., and Debreu, G.: ‘Existence of an Equilibrium for a Competitive Economy’, Econometrica 22 (1954), 265–290.

    Google Scholar 

  5. 5.

    Arrow, K. J., and Hahn, F. H.: General Competitive Analysis, Holden-Day, San Francisco, 1971.

    Google Scholar 

  6. 6.

    Artstein, Z., and Vitale, R. A.: ‘A Strong Law of Large Numbers for Random Compact Sets’, Ann. Prob. 3 (1975), 879–882.

    Google Scholar 

  7. 7.

    Aubin, J. P.: Mathematical Methods of Game and Economic Theory, North-Holland, Amsterdam, New York, Oxford, 1979.

    Google Scholar 

  8. 8.

    Aumann, R. J.: ‘Integrals of Set-Valued Functions, J. Math. Anal. Appl. 12 (1965), 1–12.

    Google Scholar 

  9. 9.

    Bagchi, S. N.: ‘On a.s. Convergence of Classes of Multivalued Asymptotic Martingales’, Abstr. Amer. Math. Soc. 5 (1984), 108–109.

    Google Scholar 

  10. 10.

    Bewley, T. F.: ‘Existence of Equilibrium in Economies with Infinitely Many Commodities’, J. Econ. Theory 4 (1972), 514–540.

    Google Scholar 

  11. 11.

    Byrne, C. L.: ‘Remarks on the Set-Valued Integrals of Debreu and Aumann’, J. Math. Anal. Appl. 62 (1978), 243–246.

    Google Scholar 

  12. 12.

    Chatterji, S. D.: ‘A Note on the Convergence of Banach-Space Valued Martingales’, Math. Ann. 153 (1964), 142–149.

    Google Scholar 

  13. 13.

    Constantinescu, C.: Spaces of Measures, DeGruyter, Berlin, New York, 1984.

    Google Scholar 

  14. 14.

    Debreu, G.: Theory of Value, Wiley, New York, London, Sydney, 1959.

    Google Scholar 

  15. 15.

    Debreu, G.: ‘Integration of Correspondences’, in Proc. Fifth Berkeley Symp. Math. Statist. Probability, Vol. II, Part 1, University of California Press, Berkeley, Los Angeles, 1967, pp. 351–372.

    Google Scholar 

  16. 16.

    Fan, K.: ‘Fixed-Point and Minimax Theorems in Locally Convex Topological Linear Spaces’, Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 121–126.

    Google Scholar 

  17. 17.

    Florenzano, M.: ‘On the Existence of Equilibria in Economies with an Infinite Dimensional Commodity Space’, J. Math. Econ. 12 (1983), 207–219.

    Google Scholar 

  18. 18.

    Ghoussoub, N., and Steele, J. M.: ‘Vector Valued Subadditive Processes and Applications in Probability’, Ann. Prob. 8 (1980), 83–95.

    Google Scholar 

  19. 19.

    Giné, E., Hahn, M. G., and Zinn, J.: ‘Limit Theorems for Random Sets: An Application of Probability in Banach Space Results’, in Probability in Banach Spaces IV (Lecture Notes in Mathematics, Vol. 990), Springer-Verlag, Berlin, Heidelberg, New York, 1983, pp. 112–135.

    Google Scholar 

  20. 20.

    Godet-Thobie, C., and Pham The, Lai: ‘Sur le plongement de l'ensemble des convexes, fermés, bornés d'un espace vectoriel topologique localement convexe dans un espace vectoriel topologique localement convexe’, C.R. Acad. Sci. Paris Série A 271 (1970), 84–87.

    Google Scholar 

  21. 21.

    Hess, C.: ‘Théorème ergodique et loi forte des grands nombres pour des ensembles aléatoires’, C.R. Acad. Sci. Paris Série A 288 (1979), 519–522.

    Google Scholar 

  22. 22.

    Hiai, F., and Umegaki, H.: ‘Integrals, Conditional Expectations, and Martingales of Multivalued Functions’, J. Multivariate Anal. 7 (1977), 149–182.

    Google Scholar 

  23. 23.

    Hörmander, L.: ‘Sur la fonction d'appui des ensembles convexes dans un espace localement convexe’, Arkiv Mat. 3 (1954), 181–186.

    Google Scholar 

  24. 24.

    Jain, N. C., and Marcus, M. B.: ‘Central Limit Theorems for C(S)-Valued Random Variables, J. Funct. Anal. 19 (1975), 216–231.

    Google Scholar 

  25. 25.

    Jarchow, H.: Locally Convex Spaces, Teubner, Stuttgart, 1981.

    Google Scholar 

  26. 26.

    Kaucher, E.: ‘Algebraische Erweiterungen der Intervallrechnung unter Erhaltung der Ordnungs- und Verbandsstrukturen’, Computing Suppl. 1 (1977), 65–79.

    Google Scholar 

  27. 27.

    Kaucher, E.: ‘Interval Analysis in the Extended Interval Space IR’, Computing Suppl. 2 (1980), 33–49.

    Google Scholar 

  28. 28.

    Kaucher, E., and Rump, S. M.: ‘E-methods for Fixed Point Equations f(x)=x’, Computing 28 (1982), 31–42.

    Google Scholar 

  29. 29.

    Kingman, J. F. C.: ‘Subadditive Ergodic Theory’, Ann. Prob. 1 (1973), 883–909.

    Google Scholar 

  30. 30.

    Klee, V. L.: ‘Some Characterizations of Reflexivity’, Rev. Ciencias (Lima) 52 (1950), 15–23.

    Google Scholar 

  31. 31.

    Krengel, U.: ‘Uh théorème ergodique pour les processus sur-stationnaires’, C.R. Acad. Sci. Paris Série A 282 (1976), 1019–1021.

    Google Scholar 

  32. 32.

    Magill, M. J. P.: ‘An Equilibrium Existence Theorem’, J. Math. Anal. Appl. 84 (1981), 162–169.

    Google Scholar 

  33. 33.

    Mourier, E.: ‘Eléments aléatoires dans un espace de Banach’, Ann. Inst. H. Poincaré 13 (1953), 161–244.

    Google Scholar 

  34. 34.

    Neveu, J.: ‘Convergence presque sûre de martingales multivoques’, Ann. Inst. H. Poincaré Section B 8 (1972), 1–7.

    Google Scholar 

  35. 35.

    Nickel, K.: ‘Verbandstheoretische Grundlagen der Intervall-Mathematik’, in Interval Mathematics (Lecture Notes in Computer Science, Vol. 29), Springer-Verlag, Berlin, Heidelberg, New York, 1975, pp. 251–262.

    Google Scholar 

  36. 36.

    Nickel, K.: ‘Intervall-Mathematik’, Z. Angew. Math. Mech. 58 (1978), T72-T85.

    Google Scholar 

  37. 37.

    Puri, M. L., and Ralescu, D. A.: ‘Strong Law of Large Numbers for Banach Space Valued Random Sets’, Ann. Prob. 11 (1983), 222–224.

    Google Scholar 

  38. 38.

    Rådström, H.: ‘An Embedding Theorem for Spaces of Convex Sets’, Proc. Amer. Math. Soc. 3 (1952), 165–169.

    Google Scholar 

  39. 39.

    Rudin, W.: Functional Analysis, McGraw-Hill, New York, 1973.

    Google Scholar 

  40. 40.

    Schaefer, H. H.: Banach Lattices and Positive Operators, Springer-Verlag, Berlin, Heidelberg, New York, 1974.

    Google Scholar 

  41. 41.

    Schauder, J.: ‘Der Fixpunktsatz in Funktionalräumen’, Studia Math. 2 (1930), 171–180.

    Google Scholar 

  42. 42.

    Schürger, K.: ‘Ergodic Theorems for Subadditive Superstationary Families of Convex Compact Random Sets’, Z. Wahrscheinlichkeitstheorie verw. Gebiete 62 (1983), 125–135.

    Google Scholar 

  43. 43.

    Tarski, A.: ‘A Lattice-Theoretical Fixpoint Theorem and its Applications’, Pacific J. Math. 5 (1955), 285–309.

    Google Scholar 

  44. 44.

    Toussaint, S.: ‘On the Existence of Equilibria in Economics with Infinitely Many Commodities and Without Ordered Preferences’, J. Econ. Theory 33 (1984), 98–115.

    Google Scholar 

  45. 45.

    Tychonoff, A.: ‘Ein Fixpunktsatz’, Math. Ann. 111 (1935), 767–776.

    Google Scholar 

  46. 46.

    Uhl jr., J. J.: ‘Applications of Radon-Nikodym Theorems to Martingale Convergence’, Trans. Amer. Math. Soc. 145 (1969), 271–285.

    Google Scholar 

  47. 47.

    Weil, W.: ‘An Application of the Central Limit Theorem for Banach-Space-Valued Random Variables to the Theory of Random Sets, Z. Wahrscheinlichkeitstheorie verw. Gebiete 60 (1982), 203–208.

    Google Scholar 

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Schmidt, K.D. Embedding theorems for classes of convex sets. Acta Appl Math 5, 209–237 (1986). https://doi.org/10.1007/BF00047343

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AMS (MOS) subject classifications (1980)

  • Primary: 52A07
  • 46A40
  • secondary: 28B20
  • 47H10
  • 54C60
  • 60D05
  • 65G10
  • 90A14

Key words

  • Classes of convex sets
  • embedding theorems
  • topological vector lattices
  • integration of random sets
  • limit theorems for random sets
  • fixed-point theorems for set-valued maps
  • mathematical economics
  • interval mathematics