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- W2896317695 abstract "Let C be a bounded convex object in ℝ d , and let P be a set of n points lying outside C . Further, let c p , c q be two integers with 1 ⩽ c q ⩽ c p ⩽ n - ⌊ d /2⌋, such that every c p + ⌊ d /2⌋ points of P contain a subset of size c q + ⌊ d /2⌋ whose convex hull is disjoint from C . Then our main theorem states the existence of a partition of P into a small number of subsets, each of whose convex hulls are disjoint from C . Our proof is constructive and implies that such a partition can be computed in polynomial time. In particular, our general theorem implies polynomial bounds for Hadwiger--Debrunner ( p , q ) numbers for balls in ℝ d . For example, it follows from our theorem that when p > q = (1+β)⋅ d /2 for β > 0, then any set of balls satisfying the ( p , q )-property can be hit by O ((1+β) 2 d 2 p 1+1/β log p ) points. This is the first improvement over a nearly 60 year-old exponential bound of roughly O (2 d ). Our results also complement the results obtained in a recent work of Keller, Smorodinsky and Tardos where, apart from improvements to the bound on HD( p , q ) for convex sets in ℝ d for various ranges of p and q , a polynomial bound is obtained for regions with low union complexity in the plane." @default.
- W2896317695 created "2018-10-26" @default.
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- W2896317695 date "2018-10-15" @default.
- W2896317695 modified "2023-10-04" @default.
- W2896317695 title "On a Problem of Danzer" @default.
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- W2896317695 doi "https://doi.org/10.1017/s0963548318000445" @default.
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