Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Helly-type theorems
for hollow axis-aligned boxes


Author: Konrad J. Swanepoel
Journal: Proc. Amer. Math. Soc. 127 (1999), 2155-2162
MSC (1991): Primary 52A35
DOI: https://doi.org/10.1090/S0002-9939-99-05220-X
Published electronically: March 3, 1999
MathSciNet review: 1646208
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A hollow axis-aligned box is the boundary of the cartesian product of $d$ compact intervals in $\mathbb{R}^d$. We show that for $d\geq 3$, if any $2^d$ of a collection of hollow axis-aligned boxes have non-empty intersection, then the whole collection has non-empty intersection; and if any $5$ of a collection of hollow axis-aligned rectangles in $\mathbb{R}^2$ have non-empty intersection, then the whole collection has non-empty intersection. The values $2^d$ for $d\geq 3$ and $5$ for $d=2$ are the best possible in general. We also characterize the collections of hollow boxes which would be counterexamples if $2^d$ were lowered to $2^d-1$, and $5$ to $4$, respectively.


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

  • 1. L. Danzer, B. Grünbaum, and V. Klee, Helly's theorem and its relatives, Convexity (V. L. Klee, ed.), Proc. of Symposia in Pure Math., vol. 7, A.M.S., 1963, pp. 100-181. MR 28:524
  • 2. M. Deza and P. Frankl, A Helly type theorem for hypersurfaces, J. Comb. Theory, Ser. A 45 (1987), 27-30. MR 88e:52012
  • 3. J. Eckhoff, Helly, Radon, and Carathéodory type theorems, Handbook of Convex Geometry (P. M. Gruber and J. M. Wills, eds.), Elsevier Science Publishers B.V., 1993, pp. 389-448. MR 94k:52010
  • 4. P. Frankl, Helly-type theorems for varieties, Europ. J. Combinatorics 10 (1989), 243-245. MR 90k:52014
  • 5. J. E. Goodman, R. Pollack, and R. Wenger, Geometric transversal theory, New Trends in Discrete and Computational Geometry (J. Pach, ed.), Springer-Verlag, Heidelberg, 1993. MR 95c:52010
  • 6. E. Helly, Über Mengen konvexer Körper mit gemeinschaftlichen Punkten, Jber. Deutsch. Math. Verein 32 (1923), 175-176.
  • 7. H. Maehara, Helly-type theorems for spheres, Discrete Comp. Geom. 4 (1989), 279-285. MR 90c:52016
  • 8. T. S. Motzkin, A proof of Hilbert's Nullstellensatz, Math. Zeitschr. 63 (1955), 341-344.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 52A35

Retrieve articles in all journals with MSC (1991): 52A35


Additional Information

Konrad J. Swanepoel
Affiliation: Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa
Email: konrad@math.up.ac.za

DOI: https://doi.org/10.1090/S0002-9939-99-05220-X
Keywords: Helly-type theorem, box, cube, hypercube
Received by editor(s): October 15, 1997
Published electronically: March 3, 1999
Communicated by: Jeffry N. Kahn
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society