On a conjecture of Danzer and Grünbaum
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- by Meir Katchalski and David Nashtir PDF
- Proc. Amer. Math. Soc. 124 (1996), 3213-3218 Request permission
Abstract:
The main result of the paper is that if $A$ is a family of homothetic triangles in the plane such that any 9 of them can be pierced by two points, then all members of $A$ can be pierced by two points. This is best possible in more than one sense: (1) the number 9 cannot be replaced by 8; (2) no similar statement is true for homothetic copies (or even translates) of a symmetric convex hexagon.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
- Ludwig Danzer and Branko Grünbaum, Intersection properties of boxes in $\textbf {R}^{d}$, Combinatorica 2 (1982), no. 3, 237–246. MR 698651, DOI 10.1007/BF02579232
- Jürgen Eckhoff, Helly, Radon, and Carathéodory type theorems, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 389–448. MR 1242986
- D. Nashtir, On a conjecture of Danzer and Grünbaum, Ms.C. Thesis, Technion, Haifa (1990), (Hebrew)
- —, Helly type problems, Ph.D. Thesis, in preparation.
Additional Information
- Meir Katchalski
- Affiliation: Department of Mathematics, Technion–Israel Institute of Technology, Haifa 32000, Israel
- Email: meirk@tx.technion.ac.il
- David Nashtir
- Affiliation: Department of Mathematics, Technion–Israel Institute of Technology, Haifa 32000, Israel
- Received by editor(s): July 18, 1994
- Additional Notes: The first author’s research was supported by the Fund for Promotion of Research at the Technion (grant 100-806) and the Technion V. P. R. Fund (grant 100-934)
- Communicated by: Jeffry N. Kahn
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 3213-3218
- MSC (1991): Primary 52A35
- DOI: https://doi.org/10.1090/S0002-9939-96-03806-3
- MathSciNet review: 1376992