TITLE: Hyperplane arrangements and covering numbers
QUESTION [5 upvotes]: Let $H$ be a set of $(d-1)$-dimensional hyperplanes in $\mathbb{R}^d$.  For each hyperplane $h \in H$ let $D(h)$ and $\bar{D}(h)$ be the corresponding half spaces of $\mathbb{R}^d$.  For a point $x \in \mathbb{R}^d$, define the function
$$
b(h,x) = \begin{cases} 
1 & x \in D(h) \\\
0 & \text{otherwise}.
\end{cases}
$$
For points $x,y \in \mathbb{R}^d$, define the psuedometric,
$$
\sigma(x,y) = \sum_{h \in H} |b(h,x) - b(h,y)|.
$$
So $\sigma(x,y)$ counts the number of hyperplanes in $H$ that pass between the points $x$ and $y$. I believe that the following lemma is true, but have not yet found a proof.
Lemma
Let $N$ be the number of hyperplanes in $H$.  There exists a polynomial $p$ (depending perhaps on $d$, but not on $N$) such that, for every positive integer $k$, there is a discrete subset $T(k) \subset \mathbb{R}^d$ containing not more than $p(k)$ elements with the property that for every $x$ in $\mathbb{R}^d$ there is a point $y \in T(k)$ with $\sigma(x,y) < \frac{N}{k}.$
Another way to state the Lemma is:  We require at most $p(k)$ `balls' of radius $\frac{N}{k}$ (with respect to the metric $\sigma$) to cover $\mathbb{R}^d$.
The Lemma is related to what is called Vapnik–Chervonenkis dimension, and it arises when trying to prove results about certain empirical processes.  I suspect this result (or something equivalent) to be in the literature, but I have not yet been able to dig it out.
Has anybody seen this result (or something equivalent) before?  Alternatively, does anybody know a nice proof?

REPLY [3 votes]: The Lemma follows from the Cutting lemma:
J. Matousek, Lectures on Discrete Geometry, Theorem 6.5.3.
This gives a partition of $\mathbb{R}^d$ into $O(k^d)$ generalized simplices (intersections of $d$ hyperplanes), each intersected by at most $n/k$ hyperplanes of $H$. To obtain the set $T(k)$, pick one point from each of the simplices.