Abstract
The Klein-Hilbert part relation, which was introduced by Gleason in function algebras and investigated for convex subsets of real vector spaces by Bear and Bauer in [3], [5], [2], is defined for convex modules. It turns out that all results that were proved for convex sets can also be proved for convex modules, which constitute the algebraic theory generated by convex sets and which have a close connection to physics and mathematical economics.
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References
Bauer, H.: An Open Mapping Theorem for Convex Sets with Only One Part,Aequ. Math. 4 (1970), 332–337.
Bauer, H. and Bear, H. S.: The Part Metric in Convex Sets,Pac. J. Math. 30 (1969), 15–33.
Bear, H. S.: A Geometric Characterisation of Gleason Parts,Proc. AMS 16 (1965), 407–412.
Bear, H. S.: Part Metric and Hyperbolic Metric,Math. Monthly (1991), 109–123.
Bear, H. S. and Weiss, M. L.: An Intrinsic Metric for Parts,Proc. AMS 18 (1967), 812–817.
Börger, R. and Kemper, R.: Cogenerators for Convex Spaces,Appl. Cat. Struct. 2 (1994), 1–11.
Bos, W. and Wolff, G.: Affine Räume I,Mitt. Math. Sem. Giessen 129 (1978), 1–115.
Bos, W. and Wolff, G.: Affine Räume II,Mitt. Math. Sem. Giessen 130 (1978), 1–83.
Dubuc, E.: Adjoint Triangles,LNM 61 (1968), 69–91.
Flood, J.: Semiconvex Geometry,J. Austral. Math. Soc. 30 (1981), 496–510.
Gudder, S. P.: Convexity and Mixtures,SIAM Rev. 19 (1977), 221–240.
Krause, U.: Strukturen in unendlichdimensionalen konvexen Mengen, Arbeitspapiere Math. 1, Univ. Bremen, 1976.
Mayer, F.: Positiv konvexe Räume, Ph.D. thesis, Fernuniversität, Hagen, 1993.
Meng, X.-W.: Categories of Convex Sets and Metric Spaces, with Applications to Stochastic Programming and Related Areas, Ph.D. thesis.
Meier, N.: Kongruenzen auf totalkonvexen Räumen, Ph.D. thesis, Fernuniversität, Hagen, 1993.
Ostermann, F. and Schmidt, J.: Der baryzentrische Kalkül als axiomatische Grundlage der affinen Geometrie,J. Reine u. Angew. Math. 224 (1966), 44–57.
Pumplün, D. and Röhrl, H.: Banach Spaces and Totally Convex Spaces I,Comm. in Alg. 12 (1985), 953–1019.
Pumplün, D. and Röhrl, H.: Banach Spaces and Totally Convex Spaces II,Comm. in Alg. 13 (1985), 1047–1113.
Pumplün, D.: Regularly Ordered Banach Spaces and Positively Convex Spaces,Res. Math. 7 (1984), 85–112.
Pumplün, D.: The Hahn-Banach Theorem for Totally Convex Spaces,Demonstr. Math. XVIII(2) (1985), 567–588.
Pumplün, D. and Röhrl, H.: Congruence Relations on Totally Convex Spaces,Comm. in Alg. 18 (1990), 1469–1496.
Röhrl, H.:Convexity Theories U — Back to the Future, Category Theory at Work, ed. by H. Herrlich and H.-E. Porst, 321–324, Heldermann, Berlin, 1991.
Röhrl, H.:Convexity Theories I — Γ-Convex Spaces, Constantin Carathéodory: An International Tribute, 1175–1209, World Sci. Publ. 1991.
Stone, M. H.: Postulates for the Barycentric Calculus,Annali di Matematica XXIX (1949), 25–30.
Swirszcz, T.: Monadic Functors and Categories of Convex Sets, Inst. Math., Polish Acad. Sci., Preprint 70, 1975.
Von Neumann, J. and Morgenstern, O.:Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1947.
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Pumplün, D., Röhrl, H. Convexity theories IV. Klein-Hilbert parts in convex modules. Appl Categor Struct 3, 173–200 (1995). https://doi.org/10.1007/BF00877635
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DOI: https://doi.org/10.1007/BF00877635