Is the cutting lemma true with O(r) lines? The cutting lemma (a.k.a. cell decomposition lemma) states that given $n$ lines in the plane it is possible to divide it into $O(r^2)$ regions (even triangles) for any $1\le r\le n$ such that the interior of any region is intersected by $O(n/r)$ lines. For more see e.g. Matousek's book Lectures on Discrete Geometry or this post.

My question is whether the plane can be divided by $O(r)$ lines (into $O(r^2)$ regions) such that the interior of any region is intersected by $O(n/r)$ of the original lines.
If the $O(r)$ lines you choose must be drawn from the original set of $n$ lines then the answer is no, it's not possible. In particular if you take $n$ lines tangent to a circle, and then choose any $r$ of them, the region containing the circle will still be intersected by all $n$ of the lines.