Optimizing for $D$ seems to yield something like $D \approx L^{-1/3} P^{2/3}$.

But then plugging in gives $$I(P,L) \ll |P|^{2/3}L^{2/3} + P.$$

This seems to prove the claim, but it is too strong. For example, consider the set of all lines through a single point.

]]>http://arxiv.org/pdf/1102.1275v1

It is a version of the so called “same type lemma”, another way of saying “geometric Szemeredi” type result.

The application there is exactly of this type, a bound from below on the so called ” space crossing number” of straight line drawings of expanders in R3. The argument also gives good bounds for the crossing number of dense or random like graphs (in R2) or 2 complexes (in R4) or any other semialgebraic relation. If the graph (complex) is not dense nor random-like then something can be said with some classic extremal combinatorics but is rather weak

Some references:

Fox, Lafforgue, Gromov, Naor and Pach used another version of the same type lemma in a very similar way, I think that they also knew the lemma for semialgebraic relations.

Pach has several versions of it in different papers with different coauthors and applications, a very nice one to a Tverberg type theorem, Barany and Valtr coined the term (and proved one) “same type lemma” . All the versions are based on Ham Sandwiching, the particular ham sandwich type result used in BH11 and in FLGNP10 is the Yao-Yao partition, Lehec has a very nice proof the Yao Yao partition (which uses only the intermediate value theorem) and Alon was the first to use it in this combinatorial context.

I wonder if one can get the Katz-Guth decomposition in one stroke with the appropriate topological statement…. ]]>