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Finite dimensional convex structures. II: The invariants. (English) Zbl 0556.52001

The paper continues the investigations started in a previous one [Topology Appl. 14, 201-225 (1982; Zbl 0506.54027)] being concerned with the invariants of Helly (\(h)\), Carathéodory (\(c)\) and Radon (\(r)\) in relation with the dimension (n) of a topological convex structure. The relations obtained \((h\leq n+1,\quad n\leq c\leq n+1)\) lead to an equivalence between Helly’s and Carathéodory’s theorem and to the closedness of the hull of compact sets in finite dimensional convexities. Also, the inequality \(r_ n\leq r\) (where \(r_ n\) is the Radon number of the n-cube) is deduced and a natural condition is presented under which \(h=c=r=n+1.\)
Reviewer: M.Turinici

MSC:

52A01 Axiomatic and generalized convexity
54H99 Connections of general topology with other structures, applications
52A35 Helly-type theorems and geometric transversal theory
54F45 Dimension theory in general topology

Citations:

Zbl 0506.54027
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References:

[1] Carathéodory, C., Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonishchen Funktionen, Rend. Circ. Mat. Palermo, 32, 193-217 (1911) · JFM 42.0429.01
[2] Eckhoff, J., Der Satz von Radon in konvexen Produktstrukturen II, Monatsh. für Math., 73, 7-30 (1969) · Zbl 0174.53701
[3] Fuchssteiner, B., Verallgemeinte Konvexitätsbegriffe und der Satz von Krein-Milman, Math. Ann., 186, 149-154 (1970) · Zbl 0184.14703
[4] de Groot, J., Topological characterization of metrizable cubes, (Theory of Sets and Topology (1972), Hausdorff Gedenkband: Hausdorff Gedenkband Berlin), 209-214 · Zbl 0262.54039
[5] Hammer, R., Beziehungen zwischen den Sätzen von Radon, Helly und Carathéodory bei axiomatischen Konvexitäten, Abh. Math. Sem. Hamburg, 46, 3-24 (1977) · Zbl 0367.52009
[6] Helly, E., Über Mengen konvekser Körper mit gemeinschaftlichen Punkten, Jber. Deutsch. Math. Verein., 32, 175-176 (1923) · JFM 49.0534.02
[7] Jamison, R. E., A general theory of convexity, (Dissertation (1974), Univ. of Washington: Univ. of Washington Seattle) · Zbl 0282.28001
[8] Jamison, R. E., Partition numbers for trees and ordered sets, Pacific J. Math., 96, 1, 115-140 (1981) · Zbl 0482.52010
[9] Kay, D. C.; Womble, E. W., Axiomatic convexity theory and relationships between the Carathéodory, Helly, and Radon numbers, Pacific J. Math., 38, 2, 471-485 (1971) · Zbl 0235.52001
[10] Klee, V., On certain intersection properties of convex sets, Canad. J. Math., 3, 272-275 (1951) · Zbl 0042.40701
[11] Lassak, M., On independent points in metric spaces, Fund. Math., 96, 53-66 (1977), (Russian) · Zbl 0374.52007
[12] Levi, F. W., On helly’s theorem and the axioms of convexity, J. Indian Math. Soc. (N.S.) A, 15, 65-76 (1951) · Zbl 0044.19101
[13] van Mill, J.; van de Vel, M., Subbases, convex sets, and hyperspaces, Pacific J. Math., 92, 2, 385-402 (1981) · Zbl 0427.54006
[15] Radon, J., Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten, Math. Ann., 83, 113-115 (1921) · JFM 48.0834.04
[16] Reay, J. R., Carathéodory theorems in convex product structures, Pacific J. Math., 35, 1, 227-230 (1970) · Zbl 0182.55601
[17] Sierksma, G., Carathéodory and Helly numbers of Convex-Product-Structures, Pacific J. Math., 61, 1, 275-282 (1975) · Zbl 0301.52002
[18] Sierksma, G., Axiomatic convexity theory and the convex product space, (Dissertation (1976), Univ. of Groningen: Univ. of Groningen Netherlands) · Zbl 0336.52001
[20] van de Vel, M., Finite dimensional convex structures I: general results, Topology Appl., 14, 201-225 (1982) · Zbl 0506.54027
[22] van de Vel, M., Two-dimensional convexities are join-hull commutative, Topology Appl., 16 (1983), to appear. · Zbl 0559.54025
[25] van de Vel, M., Matching binary convexities, Topology Appl., 16 (1983), to appear. · Zbl 0543.52001
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