Skip to main content
Log in

The generation of convex hulls

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

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

Access this article

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

Instant access to the full article PDF.

Bibliography

  1. Abe, Yoshibumi, Tomio Kubota andHajimu Yoneguchi: Some properties of a set of points in Euclidean space. Kodai Math. Seminar Reports 117–119 (1950).

  2. Banach, S.: Théorie des opérations linéaires. Warsaw 1932.

  3. Bauer, Heinz: Über die Fortsetzung positiver Linearformen. Bayer. Akad. Wiss. Math.-Nat. Kl. S.-B. 1957, 177–190 (1958).

  4. —— Silovscher Rand und Dirichletsches Problem. Ann. Inst. Fourier (Grenoble)11, 89–134 (1961).

    Google Scholar 

  5. —— Axiomatische Behandlung des Dirichletschen Problems für elliptische und parabolische Differentialgleichungen. Math. Ann.146, 1–59 (1962).

    Google Scholar 

  6. Bishop, E., andK. de Leeuw: The representation of linear functionals by measures on sets of extreme points. Ann. Inst. Fourier (Grenoble)9, 305–331 (1959).

    Google Scholar 

  7. Björck, Göran: The set of extreme points of a compact convex set. Ark. Mat.3, 463–468 (1958).

    Google Scholar 

  8. Bonnesen, F., andW. Fenchel: Theorie der konvexen Körper. Berlin: Springer 1934. (Reprint Chelsea, New York 1948.)

    Google Scholar 

  9. Bonnice, W. E.: A generalization of a theorem of Carathéodory. Am. Math. Soc. Notices8, 252–253 (1961).

    Google Scholar 

  10. Bourbaki, N.: Integration. Paris: Hermann (1952), Chaps. I–IV, A. S. I. # 1175,

    Google Scholar 

  11. Carathéodory, C.: Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen. Math. Ann.64, 95–115 (1907).

    Google Scholar 

  12. Choquet, Gustave: Theory of capacities. Ann. Inst. Fourier (Grenoble)5, 131–295 (1955).

    Google Scholar 

  13. —— Le théorème de représentation intégrale dans les ensembles convexes compacts. Ann. Inst. Fourier (Grenoble)10, 333–344 (1960).

    Google Scholar 

  14. Danzer, L., B. Grünbaum andV. Klee: Helly's theorem and its relatives. Proceedings of Symposia in Pure Mathematics Vol.7, “Convexity”, Am. Math. Soc. (1962), 101–180.

  15. Davis, Chandler: Theory of positive linear dependence. Am. J. Math.76, 733–746 (1954).

    Google Scholar 

  16. Day, Mahlon M.: Normed linear spaces. Berlin-Göttingen-Heidelberg: Springer 1958.

    Google Scholar 

  17. Dunford, Nelson, andJacob T. Schwartz: Linear operators. Part I: General theory. New York: Interscience Pub. 1958.

    Google Scholar 

  18. Fichtenholz, G., etL. Kantorovich: Sur les opérations dans l'espace des functions bornées. Studia Math.5, 69–98 (1934).

    Google Scholar 

  19. Gale, David: Linear combinations of vectors with non-negative coefficients. Am. Math. Monthly59, 46–47 (1952).

    Google Scholar 

  20. Gossieaux, Anne-Marie, etGeorges Papy: Un théorème sur les espaces du typeF. Bull. Soc. Roy. Sci. Liège13, 146–150 (1944).

    Google Scholar 

  21. Halmos, Paul R.: Measure theory. New York: Van Nostrand 1950.

    Google Scholar 

  22. Hanner, O., andH. Rådström: A generalization of a theorem of Fenchel. Proc. Am. Math. Soc.2, 589–593 (1951).

    Google Scholar 

  23. Hewitt, Edwin: Integral representation of certain linear functionals. Ark. Mat.2, 269–282 (1952).

    Google Scholar 

  24. ——, andL. J. Savage: Symmetric measures on cartesian products. Trans. Am. Math. Soc.80, 470–501 (1955).

    Google Scholar 

  25. Hildebrandt, T. H.: On bounded linear functional operations. Trans. Am. Math. Soc.36, 868–875 (1934).

    Google Scholar 

  26. Karlin, S.: Bases in Banach spaces. Duke Math. J.15, 971–985 (1948).

    Google Scholar 

  27. Klee, Victor: Extremal structure of convex sets. Arch. Math.8, 234–240 (1957).

    Google Scholar 

  28. -- The generation of affine hulls. Acta Math. Szeged24 (1963), to appear.

  29. —— Idempotency of the hull-formationH λ. Z. Wahrscheinlichkeitstheorie1, 258–262 (1963).

    Google Scholar 

  30. Krein, M., andV. Šmulian: On regularly convex sets in the space conjugate to a Banach space. Ann. Math. (2)41, 556–583 (1940).

    Google Scholar 

  31. McKinney, R. L.: Positive bases for linear spaces. Trans. Am. Math. Soc.103, 131–148 (1962).

    Google Scholar 

  32. Namioka, Isaac: Partially ordered linear topological spaces. Mem. Am. Math. Soc.24, 50 pp. (1957).

  33. Richter, Hans: Parameterfreie Abschätzung und Realisierung von Erwartungswerten. Bl. Deut. Ges. Vers.-Math.3, 147–162 (1957).

    Google Scholar 

  34. Robinson, C. V.: Spherical theorems of Helly type and congruence indices of spherical caps. Am. J. Math.64, 260–272 (1942).

    Google Scholar 

  35. Rubin, Herman, andOscar Wesler: A note on convexity in Euclideann-space. Proc. Am. Math. Soc.9, 522–523 (1958).

    Google Scholar 

  36. Sandgren, Lennart: On convex cones. Math. Scand.2, 19–28 (1954).

    Google Scholar 

  37. Schauder, J.: Über die Umkehrung linearer stetiger Funktionaloperationen. Studia Math.2, 1–6 (1930).

    Google Scholar 

  38. Steinitz, E.: Bedingt konvergente Reihen und konvexe Systeme I–II–III. J. reine angew. Math.143, 128–175 (1913);144, 1–40 (1914);146, 1–52 (1916).

    Google Scholar 

  39. Tomita, Minoru: On the regularly convex hull of a set in a conjugate Banach space. Math. J. Okayama Univ.3, 143–145 (1954).

    Google Scholar 

  40. Gurarii, V.I.: Inclinations of subspaces and conditional bases in Banach spaces. Soviet Math.3, 1028–1031 (1962).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

The work of both authors was supported in part by the National Science Foundation, USA (NSF-G18975). The paper is based on Dr.Bonnice's 1962 Ph. D. thesis at the University of Washington, written in collaboration with Prof.Klee.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bonnice, W., Klee, V.L. The generation of convex hulls. Math. Ann. 152, 1–29 (1963). https://doi.org/10.1007/BF01343728

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01343728

Keywords

Navigation