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On a problem of Danzer and Grünbaum (2014)
BibTeX
@MISC{V.14ona,
author = {Subramanya Bharadwaj B. V. and Sathish Govindarajana and Chintan Rao H},
title = {On a problem of Danzer and Grünbaum},
year = {2014}
}
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Abstract
We say that a family of subsets C of a universe U has the (l; k)-property, if every subfamily C0 C of cardinality at most l is k-hittable. In this paper we revisit a conjecture of Danzer and Grünbaum which inquires about the existence of a function g(k; d) such that if any family of objects C in Rd has the (g(k; d); k)-property, then C is k-piercable (hittable). Danzer and Grünbaum showed that g(k; d) is infinite for families of boxes and translates of centrally symmetric convex hexagons. Negative results for other families of objects have been shown. Very few positive results are known and only for small values of k. In this paper, we show that, for a family of subsets C of a given universe U, with the property that any subfamily of p+1 distinct subsets in C intersect in at most l elements, if C has the ((k(klp+1)+ 1; k)-property then C is k-hittable. We use the above result to show Danzer Grünbaum type results for pseudolines and halfplanes in Rd. These are the first positive results for arbitrary k in a general setting. We also pose a relaxed version of the above question which we call the k-helly problem: For every positive integer k, determine the smallest f(k; d) such that if any family of convex objects C in Rd has the (g(k; d); k)-property for some g(k; d), then C is f(k; d)-piercable. We show that any family of boxes in Rd with the (k2d; k)-property is 2dk-piercable. We finally show the existence of polynomial sized kernels for the corresponding hitting set and covering problems in the setting described above.
