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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] Branko Grünbaum, The dimension of intersections of convex sets, Pacific J. Math. 12 (1962), 197–202. MR 0142054
  • [4] Hugo Hadwiger and Hans Debrunner, Combinatorial geometry in the plane, Translated by Victor Klee. With a new chapter and other additional material supplied by the translator, Holt, Rinehart and Winston, New York, 1964. MR 0164279
  • [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 University Press, London-New York, 1971. Prepared in collaboration with J. E. Reeve and A. A. Ball; London Mathematical Society Lecture Note Series, 3. MR 0301635
  • [7] Johann Radon, Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten, Math. Ann. 83 (1921), no. 1-2, 113–115 (German). MR 1512002, https://doi.org/10.1007/BF01464231
  • [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] Fritz John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, N. Y., 1948, pp. 187–204. MR 0030135

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Additional Information

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

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