Radon’s theorem revisited

  • Jürgen Eckhoff


Radon’s theorem is one of the cornerstones of combinatorial geometry. It asserts that each set of d + 2 points in Rd can be expressed as the union of two disjoint subsets whose convex hulls have a common point. Moreover, the number d + 2 is the smallest which has the stated property.


Convex Hull General Position Convexity Space Convex Polytope Collinear Point 
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  1. E.G. Bajmóczy, I. Bârâny, (1977) On a common generalization of Borsuk’s and Radon’s theorem. To appearGoogle Scholar
  2. I. Bârâny, S.B. Shlosman, A. Szücs, (1978) On a topological generalization of a theorem of Tverberg. To appear.Google Scholar
  3. P.W. Bean, (1974) Helly and Radon-type theorems in interval convexity spaces. Pacific J. Math. 51 (1974), 363–368.Google Scholar
  4. P.A. Beck, (1945) Determination of coexisting phases in heterogeneous systems of many components. J. Appl. Phys. 16 (1945), 808–815.CrossRefGoogle Scholar
  5. B.J. Birch, (1959) On 3N points in a plane. Proc. Cambridge Philos. Soc. 55 (1959), 289–293.Google Scholar
  6. M. Breen, (1970) A determination of the combinatorial type of a polytope by Radon partitions. Ph. D. Thesis, Clemson University (1970).Google Scholar
  7. M. Breen, (1972) Determining a polytope by Radon partitions. Pacific J. Math. 43 (1972), 27–37.Google Scholar
  8. M. Breen, (1973) Primitive Radon partitions for cyclic polytopes. Israel J. Math. 15 (1973), 156–157.Google Scholar
  9. B. Brosowski, (1965) Über Extremalsignaturen linearer Polynome in n Veränderlichen. Numer. Math. 7 (1965), 396–405.CrossRefGoogle Scholar
  10. V.W. Bryant, R.J. Webster, (1969) Generalizations of the theorems of Radon, Helly, and Carathéodory. Monatsh. Math. 73 (1969), 309–315.CrossRefGoogle Scholar
  11. T.H. Brylawski, (1976) A combinatorial perspective on the Radon convexity theorem. Geometriae Dedicata 5 (1976), 459–466.Google Scholar
  12. T.H. Brylawski, (1977) Connected matroids with the smallest Whitney numbers. Discrete Math. 18 (1977), 243252.Google Scholar
  13. J.R. Calder, (1971) Some elementary properties of interval convexities. J. London Math. Soc. (2) 3 (1971), 422428.Google Scholar
  14. J. Cantwell, (1974) Geometric convexity. I. Bull. Inst. Math. Acad. Sinica 2 (1974), 289–307.Google Scholar
  15. L. Danzer, B. Grünbaum, V. Klee, (1963) Helly’s theorem and its relatives. Proc. Sympos. Pure Math. 7 (Convexity), pp. 101–180. Amer. Math. Soc., Providence 1963.Google Scholar
  16. R. De Santis, (1957) A generalization of Helly’s theorem. Proc. Amer. Math. Soc. 8 (1957), 336–340.Google Scholar
  17. J.-P. Doignon, (1974/75) Segments et ensembles convexes. Thèse, Université Libre de Bruxelles (1974/75).Google Scholar
  18. J.-P. Doignon, (1978) Radon partitions with k-dimensional intersection. To appear.Google Scholar
  19. J.-P. Doignon, (1979) Quelques problèmes de convexité combinatoire. Bull. Soc. Math. Belg. To appear. J.-P. Doignon, J.R. Reay, G. Sierksma, ( 1979 ) A Tverberg-type generalization of the Helly number of a convexity space. In preparation.Google Scholar
  20. J.-P. Doignon, G. Valette, ( 1975 ) Variations sur un thème de Radon. Notes polycopiées, Université Libre de Bruxelles (1975).Google Scholar
  21. J.-P. Doignon, G. Valette, (1977) Radon partitions and a new notion of independence in affine and projective spaces. Mathematika 24 (1977), 86–96.CrossRefGoogle Scholar
  22. A. Dowling, R.M. Wilson, (1974) The slimmest geometric lattices. Trans. Amer. Math. Soc. 196 (1974), 203–215.CrossRefGoogle Scholar
  23. J. Eckhoff, (1968) Der Satz von Radon in konvexen Produktstrukturen I. Monatsh. Math. 72 (1968), 303–314.CrossRefGoogle Scholar
  24. J. Eckhoff, (1969) Der Satz von Radon in konvexen Produktstrukturen II. Monatsh. Math. 73 (1969), 7–30.CrossRefGoogle Scholar
  25. J. Eckhoff, (1974) Primitive Radon partitions. Mathematika 21 (1974), 32–37.CrossRefGoogle Scholar
  26. J. Eckhoff, (1975) Radonpartitionen und konvexe Polyeder. J. Reine Angew. Math. 277 (1975), 120–129.Google Scholar
  27. J. Eckhoff, (1976) On a class of convex polytopes. Israel J. Math. 23 (1976), 332–336.Google Scholar
  28. B. Grünbaum, (1967) Convex Polytopes. Wiley and Sons, London, New York, Sydney 1967.Google Scholar
  29. R. Hammer, (1977) Beziehungen zwischen den Sätzen von Radon, Helly und Carathéodory bei axiomatischen Konvexitäten. Abh. Math. Sem. Univ. Hamburg 46 (1977), 3–24.CrossRefGoogle Scholar
  30. W. Hansen, V. Klee, (1969) Intersection theorems for positive sets. Proc. Amer. Math. Soc. 22 (1969), 450–457.CrossRefGoogle Scholar
  31. W.R. Hare, J.W. Kenelly, (1971) Characterizations of Radon partitions. Pacific J. Math. 36 (1971), 159–164.Google Scholar
  32. W.R. Hare, G. Thompson, (1975) Tverberg-type theorems in convex product structures. Geom. metric lin. Spaces, Proc. Conf. East Lansing 1974 (ed. L.M. Kelly). Lecture Notes Math. 490 (1975), 212–217.Google Scholar
  33. L.M. Iversland, (1969) On convex sets in R. (Norwegian). Cand. Real. Thesis, University of Bergen (1969).Google Scholar
  34. R.E. Jamison, (1979) Partition numbers of trees and ordered sets. Preprint, Technische Hochschule Darmstadt (1979).Google Scholar
  35. D.C. Kay, E.W. Womble, (1971) Axiomatic convexity theory and relationships between the Carathéodory, Helly, and Radon numbers. Pacific J. Math. 38 (1971), 471–485.Google Scholar
  36. N.N. Khudekov, (1941) Über die allgemeine Lage von n+2 Punkten in R“. (Russian, German summary). Mat. Sbornik N.S. 9 (51) (1941), 249–276.Google Scholar
  37. B. Kind, P. Kleinschmidt, (1976) On the maximal volume of convex bodies with few vertices. J. Combinatorial Theory (A) 21 (1976), 124–128.CrossRefGoogle Scholar
  38. L. Kosmâk, (1963) A remark on Helly’s theorem. (Czech. Russian and English Summaries). Spisy Prirod. Fak. Univ. Brné 1963, 223–225.Google Scholar
  39. D. Kramer, (1975) Lineare Relationen zwischen Radonvektoren, primitiven Radonvektoren und f-Vektoren. Manuskript, Universität Dortmund (1975).Google Scholar
  40. D.G. Larman, (1972) On sets projectively equivalent to the vertices of a convex polytope. Bull. London Math.Soc. 4 (1972), 6–12.CrossRefGoogle Scholar
  41. F.W. Levi, (1951) On Helly’s theorem and the axioms of convexity. J. Indian Math. Soc. (N.S.) 15 (1951), 65–76.Google Scholar
  42. A. Markoff, (1939) On the definition of a complex. (English, Russian summary). Mat. Sbornik N.S. 5 (47) (1939), 545–550.Google Scholar
  43. P. McMullen, D.W. Walkup, (1971) A generalized lower-bound conjecture for simplicial polytopes. Mathematika 18 (1971), 264–273.CrossRefGoogle Scholar
  44. P.M. Pepper (1949) Coexistence relations of n + 1 phases in n-component systems. J. Appl. Phys. 20 (1949), 754–760.CrossRefGoogle Scholar
  45. B. B. Peterson, (1972) The geometry of Radon’s theorem. Amer. Math. Monthly 79 (1972), 949–963.CrossRefGoogle Scholar
  46. C.M. Petty, (1975) Radon partitions in real linear spaces. Pacific J. Math. 59 (1975), 513–523.Google Scholar
  47. I.V. Proskuryakov, (1959) A property of n-dimensional affine space connected with Helly’s theorem. (Russian). Uspehi Mat. Nauk 14 (1959), No. 1 (85), 219–222.Google Scholar
  48. R. Rado, (1952) Theorems on the intersection of convex sets of points. J. London Math. Soc. 27 (1952), 320–328.CrossRefGoogle Scholar
  49. J. Radon, (1921) Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten. Math. Ann. 83 (1921), 113–115.CrossRefGoogle Scholar
  50. J.R. Reay, (1968) An extension of Radon’s theorem. Illinois J. Math. 12 (1968), 184–189.Google Scholar
  51. J.R. Reay, (1972) Convex hulls of points and directions. Notices Amer. Math. Soc. 19 (1972), A-192.Google Scholar
  52. J.R. Reay, (1978) Several generalizations of Tverberg’s theorem. Israel J. Math. To appear. (See also: Radon type theorems without independence conditions. Notices Amer. Math. Soc. 23 (1976), A-540.)Google Scholar
  53. J.R. Reay, (1979) Twelve general position points always form three intersecting tetrahedra. Technical Report,Western Washington University.Google Scholar
  54. J.R. Reay, P.W. Bean, (1974) Radon-type theorems in partial order convexity spaces. Notices Amer. Math. Soc. 21(1974), A-204.Google Scholar
  55. L.G. Saraburova, Ju.A. Saskin, (1975) Intersections of spherically convex sets. (Russian). Mat. Zametki 18 (1975), 781–791.(English transl.: Math. Notes 18 (1975), 1054–1059.CrossRefGoogle Scholar
  56. Ju.A. Saskin, (1967) Interpolation families of functions and imbedding of sets into euclidean and projective spaces. (Russian). Dokl. Akad. Nauk SSSR 174 (1967), 1030–1032. (English transi.: Soviet Math. Dokl. 8 (1967), 722–725.)Google Scholar
  57. G.C. Shephard, (1969) Neighbourliness and Radon’s theorem. Mathematika 16 (1969), 273–275.CrossRefGoogle Scholar
  58. G. Sierksma, (1976) Axiomatic convexity theory and the convex product space. Ph. D. Thesis, University of Groningen (1976).Google Scholar
  59. G. Sierksma, (1977) Relationships between Carathéodory, Helly, Radon and exchange numbers of convexity spaces. Nieuw Arch. Wisk. (3) 25 (1977), 115–132.Google Scholar
  60. G. Sierksma, J.Ch. Boland, (1974) The least-upper-bound for the Radonnumber of an Eckhoff-space. Report, University of Groningen (1974).Google Scholar
  61. J. Stangeland, (1978) Convex properties of finite sequences of points in R. (Norwegian). Cand. Real. Thesis, University of Bergen (1978).Google Scholar
  62. H. Tverberg, (1966) A generalization of Radon’s theorem. J. London Math. Soc. 41 (1966), 123–128.CrossRefGoogle Scholar
  63. H. Tverberg, (1968) A further generalization of Radon’s theorem. J. London Math. Soc. 43 (1968), 352–354.CrossRefGoogle Scholar
  64. H. Tverberg, (1971) On equal unions of sets. Studies in Pure Mathematics, Papers presented to Richard Rado (ed. L. Mirsky), pp. 249–250. Academic Press, London, New York 1971.Google Scholar
  65. D.J.A. Welsh, (1976) Matroid Theory. Academic Press, London, New York, San Francisco 1976. R.O. Winder, (1966) Partitions of n-space by hyperplanes. SIAM J. Appl. Math. 14 (1966), 811–818.Google Scholar
  66. Th. Zaslaysky, (1975) Facing up to arrangements: face-count formulas for partitions of space by hyperplanes. Mem. Amer. Math. Soc. 1 (1975), No. 154, vii+102 pp.Google Scholar

Copyright information

© Springer Basel AG 1979

Authors and Affiliations

  • Jürgen Eckhoff
    • 1
  1. 1.Abteilung MathematikUniversität DortmundDortmund 5oDeutschland

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