Helly-type theorems for hollow axis-aligned boxes
HTML articles powered by AMS MathViewer
- by Konrad J. Swanepoel PDF
- Proc. Amer. Math. Soc. 127 (1999), 2155-2162 Request permission
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
- Ludwig Danzer, Branko Grünbaum, and Victor Klee, Helly’s theorem and its relatives, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 101–180. MR 0157289
- M. Deza and P. Frankl, A Helly type theorem for hypersurfaces, J. Combin. Theory Ser. A 45 (1987), no. 1, 27–30. MR 883890, DOI 10.1016/0097-3165(87)90043-4
- Jürgen Eckhoff, Helly, Radon, and Carathéodory type theorems, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 389–448. MR 1242986
- P. Frankl, Helly-type theorems for varieties, European J. Combin. 10 (1989), no. 3, 243–245. MR 1029170, DOI 10.1016/S0195-6698(89)80058-7
- Jacob E. Goodman, Richard Pollack, and Rephael Wenger, Geometric transversal theory, New trends in discrete and computational geometry, Algorithms Combin., vol. 10, Springer, Berlin, 1993, pp. 163–198. MR 1228043, DOI 10.1007/978-3-642-58043-7_{8}
- E. Helly, Über Mengen konvexer Körper mit gemeinschaftlichen Punkten, Jber. Deutsch. Math. Verein 32 (1923), 175–176.
- Hiroshi Maehara, Helly-type theorems for spheres, Discrete Comput. Geom. 4 (1989), no. 3, 279–285. MR 988750, DOI 10.1007/BF02187730
- T. S. Motzkin, A proof of Hilbert’s Nullstellensatz, Math. Zeitschr. 63 (1955), 341–344.
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
- Received by editor(s): October 15, 1997
- Published electronically: March 3, 1999
- Communicated by: Jeffry N. Kahn
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 2155-2162
- MSC (1991): Primary 52A35
- DOI: https://doi.org/10.1090/S0002-9939-99-05220-X
- MathSciNet review: 1646208