Proceedings of the American Mathematical Society

Author(s): Meir Katchalski; David Nashtir
Journal: Proc. Amer. Math. Soc. 124 (1996), 3213-3218.
MSC (1991): Primary 52A35
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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.

1.: L. Danzer, B. Grünbaum and V. Klee, Helly's theorem and its relatives, Proc. Symposia in Pure Math., vol. VII (Convexity) (1963), 101--180. MR 28:524
2.: L. Danzer and B. Grünbaum, Intersection properties of boxes in , Combinatorica 2 (1982), 237--246. MR 84g:52014
3.: J. Eckhoff, Helly Radon and Carathéodory Type Theorems, Chapter 2.1 in P. M. Gruber and J. M. Wills (eds.), Handbook of Convex Geometry, North-Holland (1993), 389--448. MR 94k:52010
4.: D. Nashtir, On a conjecture of Danzer and Grünbaum, Ms.C. Thesis, Technion, Haifa (1990), (Hebrew)
5.: ------, Helly type problems, Ph.D. Thesis, in preparation.

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

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

DOI: 10.1090/S0002-9939-96-03806-3
PII: S 0002-9939(96)03806-3
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 of article: Copyright 1996, American Mathematical Society

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