In the paper "On the Erdős distinct distance problem in the plane" by Larry Guth and Nets Hawk Katz,


<http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.4105v1.pdf>


they define the functions $d(P)$, (distinct distances between $N$ points), and $Q(P)$, (quadruples on $N$ points where the distances between the first two points of the quadruple are equal to the second two points). They then apply Cauchy Schwarz to obtain


$|d(P)|\ge\frac{N^4}{|Q(P)|}$


However, I don't see how we can consider the functions $d$ and $Q$ to be functions in the same Vector Space, since one yields a set of distances, where the other yields a set of quadruples. If I can't interpret them as elements of a Vector Space, I don't see how Cauchy Schwarz can be applied.



Determining the vector space for application of Cauchy Schwarz
Let $d(P)=\{d\_1,\dots,d\_r\}$ and write $P^2=\cup\_{i=1}^r S\_i$, where $(p\_1,p\_2)\in S\_i$ if $|p\_1,p\_2|=d\_i$. Now you can see that $Q(P)=\cup (S\_i\times S\_i)$ and so the inequality is only saying
$$r(s\_1^2+\cdots+s\_r^2)\geq (s\_1+\cdots+s\_r)^2$$
because $s\_1+\cdots+s\_r=N^2$.


