Finding Points in Convex Position in Density-Restricted Sets
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For a finite set A⊂ℝ^d, let Δ(A) denote the spread of A, which is the ratio of the maximum pairwise distance to the minimum pairwise distance. For a positive integer n, let γ_d(n) denote the largest integer such that any set A of n points in general position in ℝ^d, satisfying Δ(A) ≤α n^1/d for a fixed α>0, contains at least γ_d(n) points in convex position. About 30 years ago, Valtr proved that γ_2(n)=Θ(n^1/3). Since then no further results have been obtained in higher dimensions. Here we continue this line of research in three dimensions and prove that γ_3(n) =Θ(n^1/2). The lower bound implies the following approximation: Given any n-element point set A⊂ℝ^3 in general position, satisfying Δ(A) ≤α n^1/3 for a fixed α, a Ω(n^-1/6)-factor approximation of the maximum-size convex subset of points can be computed by a randomized algorithm in O(n logn) expected time.
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