  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).
