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Quantitative Helly-type theorems


Authors: Imre Bárány, Meir Katchalski and János Pach
Journal: Proc. Amer. Math. Soc. 86 (1982), 109-114
MSC: Primary 52A35
DOI: https://doi.org/10.1090/S0002-9939-1982-0663877-X
MathSciNet review: 663877
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Abstract: We establish some quantitative versions of Helly's famous theorem on convex sets in Euclidean space. We prove, for instance, that if $ \mathcal{C}$ is any finite family of convex sets in $ {{\mathbf{R}}^d}$, such that the intersection of any $ 2d$ members of $ \mathcal{C}$ has volume at least 1, then the intersection of all members belonging to $ \mathcal{C}$ is of volume $ \geqslant {d^{ - {d^2}}}$. A similar theorem is true for diameter, instead of volume. A quantitative version of Steinitz' Theorem is also proved.


References [Enhancements On Off] (What's this?)

  • [1] I. Bárány, M. Katchalski and J. Pach, Helly's theorem, Amer. Math. Monthly (submitted).
  • [2] L. Danzer, B. Grünbaum and V. Klee, Helly's Theorem and its relatives, Proc. Sympos. Pure Math., vol. 7, Amer. Math. Soc., Providence, R. I., 1963, pp. 101-180.
  • [3] B. Grünbaum, The dimensions of intersections of convex sets, Pacific J. Math. 12 (1962), 197-202. MR 0142054 (25:5448)
  • [4] H. Hadwiger, H. Debrunner and V. Klee, Combinational geometry in the plane, Holt, Rinehart & Winston, New York, 1964. MR 0164279 (29:1577)
  • [5] E. Helly, Über Mengen konvexer Körper mit gemeinschaftlichen Punkten, Jber. Deutsch. Math.-Verein. 32 (1923), 175-176.
  • [6] P. McMullen and G. C. Shephard, Convex polytopes and the upper bound conjecture, Cambridge Univ. Press, New York, 1971. MR 0301635 (46:791)
  • [7] J. Radon, Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten, Math. Ann. 83 (1921), 113-115. MR 1512002
  • [8] E. Steinitz, Bedingt konvergente Reihen und konvexe Systemes. I-II-III, J. Reine Angew. Math. 143 (1913), 128-175, 144 (1914), 1-40, 146 (1916), 1-52.
  • [9] F. John, Extremum problems with inequalities as subsidiary conditions, Courant Anniversary Volume, Interscience, New York, 1948, pp. 187-204. MR 0030135 (10:719b)

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DOI: https://doi.org/10.1090/S0002-9939-1982-0663877-X
Article copyright: © Copyright 1982 American Mathematical Society

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