Necessary and sufficient conditions for hyperplane transversals

Abstract

We prove that a finite family ℬ={B 1,B 2, ...,B n } of connected compact sets in ℝd 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 ℝk and everyk+2 sets of ℬ 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.

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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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Pollack, R., Wenger, R. Necessary and sufficient conditions for hyperplane transversals. Combinatorica 10, 307–311 (1990). https://doi.org/10.1007/BF02122783

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AMS subject classification (1980)

  • 52 A 35
  • 52 A 20