Bibliography
Abe, Yoshibumi, Tomio Kubota andHajimu Yoneguchi: Some properties of a set of points in Euclidean space. Kodai Math. Seminar Reports 117–119 (1950).
Banach, S.: Théorie des opérations linéaires. Warsaw 1932.
Bauer, Heinz: Über die Fortsetzung positiver Linearformen. Bayer. Akad. Wiss. Math.-Nat. Kl. S.-B. 1957, 177–190 (1958).
—— Silovscher Rand und Dirichletsches Problem. Ann. Inst. Fourier (Grenoble)11, 89–134 (1961).
—— Axiomatische Behandlung des Dirichletschen Problems für elliptische und parabolische Differentialgleichungen. Math. Ann.146, 1–59 (1962).
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).
Björck, Göran: The set of extreme points of a compact convex set. Ark. Mat.3, 463–468 (1958).
Bonnesen, F., andW. Fenchel: Theorie der konvexen Körper. Berlin: Springer 1934. (Reprint Chelsea, New York 1948.)
Bonnice, W. E.: A generalization of a theorem of Carathéodory. Am. Math. Soc. Notices8, 252–253 (1961).
Bourbaki, N.: Integration. Paris: Hermann (1952), Chaps. I–IV, A. S. I. # 1175,
Carathéodory, C.: Über den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen. Math. Ann.64, 95–115 (1907).
Choquet, Gustave: Theory of capacities. Ann. Inst. Fourier (Grenoble)5, 131–295 (1955).
—— Le théorème de représentation intégrale dans les ensembles convexes compacts. Ann. Inst. Fourier (Grenoble)10, 333–344 (1960).
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.
Davis, Chandler: Theory of positive linear dependence. Am. J. Math.76, 733–746 (1954).
Day, Mahlon M.: Normed linear spaces. Berlin-Göttingen-Heidelberg: Springer 1958.
Dunford, Nelson, andJacob T. Schwartz: Linear operators. Part I: General theory. New York: Interscience Pub. 1958.
Fichtenholz, G., etL. Kantorovich: Sur les opérations dans l'espace des functions bornées. Studia Math.5, 69–98 (1934).
Gale, David: Linear combinations of vectors with non-negative coefficients. Am. Math. Monthly59, 46–47 (1952).
Gossieaux, Anne-Marie, etGeorges Papy: Un théorème sur les espaces du typeF. Bull. Soc. Roy. Sci. Liège13, 146–150 (1944).
Halmos, Paul R.: Measure theory. New York: Van Nostrand 1950.
Hanner, O., andH. Rådström: A generalization of a theorem of Fenchel. Proc. Am. Math. Soc.2, 589–593 (1951).
Hewitt, Edwin: Integral representation of certain linear functionals. Ark. Mat.2, 269–282 (1952).
——, andL. J. Savage: Symmetric measures on cartesian products. Trans. Am. Math. Soc.80, 470–501 (1955).
Hildebrandt, T. H.: On bounded linear functional operations. Trans. Am. Math. Soc.36, 868–875 (1934).
Karlin, S.: Bases in Banach spaces. Duke Math. J.15, 971–985 (1948).
Klee, Victor: Extremal structure of convex sets. Arch. Math.8, 234–240 (1957).
-- The generation of affine hulls. Acta Math. Szeged24 (1963), to appear.
—— Idempotency of the hull-formationH λ. Z. Wahrscheinlichkeitstheorie1, 258–262 (1963).
Krein, M., andV. Šmulian: On regularly convex sets in the space conjugate to a Banach space. Ann. Math. (2)41, 556–583 (1940).
McKinney, R. L.: Positive bases for linear spaces. Trans. Am. Math. Soc.103, 131–148 (1962).
Namioka, Isaac: Partially ordered linear topological spaces. Mem. Am. Math. Soc.24, 50 pp. (1957).
Richter, Hans: Parameterfreie Abschätzung und Realisierung von Erwartungswerten. Bl. Deut. Ges. Vers.-Math.3, 147–162 (1957).
Robinson, C. V.: Spherical theorems of Helly type and congruence indices of spherical caps. Am. J. Math.64, 260–272 (1942).
Rubin, Herman, andOscar Wesler: A note on convexity in Euclideann-space. Proc. Am. Math. Soc.9, 522–523 (1958).
Sandgren, Lennart: On convex cones. Math. Scand.2, 19–28 (1954).
Schauder, J.: Über die Umkehrung linearer stetiger Funktionaloperationen. Studia Math.2, 1–6 (1930).
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).
Tomita, Minoru: On the regularly convex hull of a set in a conjugate Banach space. Math. J. Okayama Univ.3, 143–145 (1954).
Gurarii, V.I.: Inclinations of subspaces and conditional bases in Banach spaces. Soviet Math.3, 1028–1031 (1962).
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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.
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Bonnice, W., Klee, V.L. The generation of convex hulls. Math. Ann. 152, 1–29 (1963). https://doi.org/10.1007/BF01343728
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DOI: https://doi.org/10.1007/BF01343728