Chirotopes of Random Points in Space are Realizable on a Small Integer Grid
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We prove that with high probability, a uniform sample of n points in a convex domain in ℝ^d can be rounded to points on a grid of step size proportional to 1/n^d+1+ϵ without changing the underlying chirotope (oriented matroid). Therefore, chirotopes of random point sets can be encoded with O(nlog n) bits. This is in stark contrast to the worst case, where the grid may be forced to have step size 1/2^2^Ω(n) even for d=2. This result is a high-dimensional generalization of previous results on order types of random planar point sets due to Fabila-Monroy and Huemer (2017) and Devillers, Duchon, Glisse, and Goaoc (2018).
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