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!



Cutting Lemma in Discrete Geometry

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> Chazelle B, "Cuttings." *Handbook of Data Structures and Applications* (D. Mehta and S. Sahni, editors), chap. 25. 2005. [PDF download](https://www.cs.princeton.edu/~chazelle/pubs/cuttings.pdf).
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**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.


