Necessary and sufficient conditions for hyperplane transversals |
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Authors: | R Pollack R Wenger |
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Institution: | (1) Courant Institute of Mathematical Science, New York University, USA;(2) Center for Discrete Mathematics and Theoretical Computer Science, Hill Center for the Mathematical Sciences, Rutgers University, 08903 New Brunswick, NJ, USA |
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Abstract: | We prove that a finite family ={B
1,B
2, ...,B
n
} of connected compact sets in
d
has a hyperplane transversal if and only if for somek there exists a set of pointsP={p
1,p
2, ...,p
n
} (i.e., ak-dimensional labeling of the family) which spans
k
and everyk+2 sets of are met by ak-flat consistent with the order type ofP. This is a common generalization of theorems of Hadwiger, Katchalski, Goodman-Pollack and Wenger.Supported in part by NSF grant DMS-8501947 and CCR-8901484, NSA grant MDA904-89-H-2030, and the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), a National Science Foundation Science and Technology Center, under NSF grant STC88-09648.Supported by the National Science and Engineering Research Council of Canada and DIMACS. |
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Keywords: | 52 A 35 52 A 20 |
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