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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 

 

Piercing convex sets


Authors: Noga Alon and Daniel J. Kleitman
Journal: Bull. Amer. Math. Soc. 27 (1992), 252-256
MSC (2000): Primary 52A35
MathSciNet review: 1149871
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Abstract: A family of sets has the $ (p,q)$ property if among any p members of the family some q have a nonempty intersection. It is shown that for every $ p \geq q \geq d + 1$ there is a $ c = c(p,q,d) < \infty $ such that for every family $ \mathcal{F}$ of compact, convex sets in $ {R^d}$ that has the ($ (p,q)$) property there is a set of at most c points in $ {R^d}$ that intersects each member of $ \mathcal{F}$. This extends Helly's Theorem and settles an old problem of Hadwiger and Debrunner.


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DOI: https://doi.org/10.1090/S0273-0979-1992-00304-X
Article copyright: © Copyright 1992 American Mathematical Society