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arXiv · 2001.08062

Chirotopes of Random Points in Space are Realizable on a Small Integer Grid

Jean Cardinal·Ruy Fabila-Monroy·Carlos Hidalgo-Toscano
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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).

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Jean Cardinal, Ruy Fabila-Monroy, Carlos Hidalgo-Toscano. 2020-01-22. Chirotopes of Random Points in Space are Realizable on a Small Integer Grid. https://arxiv.org/abs/2001.08062

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