# Cutting Lemma in Discrete Geometry

I'm looking for a survey or a source for Cutting Lemma. I looked at Matusek's Discrete Geometry textbook, but it only proved Cutting Lemma for lines in $\mathbb{R}^2.$ I need to know the proof in $\mathbb{R}^d$ and other variations of Cutting Lemma.

Thanks in advance for any help!

Theorem 1.1. Given a set $H$ of $n$ hyperplanes in $\mathbb{R}^d$, for any $0 < \epsilon < 1$, there exists an $\epsilon$-cutting for $H$ of size $O(\epsilon^{-d})$, which is optimal. The cutting, together with the list of hyperplanes intersecting the interior of each simplex, can be found deterministically in $O(n \epsilon^{1-d})$ time.