Suppose we are given a disc of measure 1. What is the smallest transversal for the set of halfplanes that contain 1% of it?
(Another related puzzle: Connect 3x3 grid with polygonal line that consists of 4 segments.)
Weak epsilon net - makes sense if measure is uniform on a finite point set and sets are all convex sets.
Weak epsilon net thm: For any n points and r, there is a weak 1/r-net of size f(d,r)=O(r^d poly_d (log r)).
Proof: We only prove r^{d+1}, using the first selection lemma.
Until there is set of size n/r not stabbed, we can select a point in many of its simplices.
We repeat this until such a set exists, at most Cr^{d+1} times.
Corollary: For F family of convex sets, tau(F)<=f(d,tau*).
Proof: Instead of uniform measure on points, we can take measure given by tau*, then like eps-net thm.
In plane r poly(log r) is known and recently a construction with super-linear r was found by Alon, using:
Hales-Jewett thm: For every k there is a d such that if we two-color [k]^d, then there is a line from one of the colors.
Corollary: van der Waerden thm: If we two-color the integers, there is arbitrarily long arithmetic progression.
Density version [Szemeredi]: If we select 1% of integers, there is arbitrarily long arithmetic progression.
Density HJ thm: For any k and p there is a d such that if we select p fraction of points of [k]^d, there is a line.
Moreover, there is a combinatorial line, i.e. one with an "all-positive" slope.
For a recent combinatorial proof see the Polymath1 project.