Colorful Helly Theorem for Piercing Boxes with Two Points
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Let h(d,2) denote the smallest integer such that any finite collection of axis parallel boxes in ℝ^d is two-pierceable if and only if every h(d,2) many boxes from the collection is two-pierceable. Danzer and Grünbaum (1982) proved that h(d,2) equals 3d for odd d and (3d-1) for even d. In this short note paper, we have given an optimal colorful generalization of the above result, and using it derived a new fractional Helly Theorem for two-piercing boxes in ℝ^d. We have also shown that using our fractional Helly theorem and following the same techniques used by Chakraborty et al. (2018) we can design a constant query algorithm for testing if a set of points is (2,G)-clusterable, where G is an axis parallel box in ℝ^d.
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