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Chirotopes of Random Points in Space are Realizable on a Small Integer Grid (2001.08062v1)
Published 22 Jan 2020 in cs.CG
Abstract: We prove that with high probability, a uniform sample of $n$ points in a convex domain in $\mathbb{R}d$ can be rounded to points on a grid of step size proportional to $1/n{d+1+\epsilon}$ without changing the underlying chirotope (oriented matroid). Therefore, chirotopes of random point sets can be encoded with $O(n\log n)$ bits. This is in stark contrast to the worst case, where the grid may be forced to have step size $1/2{2{\Omega(n)}}$ 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).