I've occasionally heard it stated (most notably on Terry Tao's blog) that "the Cauchy-Schwarz inequality can be viewed as a quantitative strengthening of the pigeonhole principle." I've certainly seen the inequality put to good use, but I haven't seen anything to make me believe that statement on the same level that I believe that the probabilistic method can be used as a (vast) strengthening of pigeonhole.
So, how exactly can Cauchy-Schwarz be seen as a quantitative version of the pigeonhole principle? And for extra pigeonholey goodness, are there similarly powered-up versions of the principle's other generalizations? (Linear algebra arguments [particularly dimension arguments], the probabilistic method, etc.)
Best Answer
My own interpretation (which I guess is pretty similar to the one above):
Suppose you have $r$ pigeons and $n$ holes, and want to minimize the number of pairs of pigeons in the same hole. This can easily be seen as equivalent to minimizing the sum of the squares of the number of pigeons in each hole.
Classical Cauchy Schwarz: $x_1^2+...+x_n^2 \ge\displaystyle\frac1n(x_1+...+x_n)^2$.
Discrete Cauchy Schwarz: If you must place an integer number of pigeons in each hole, the number of pairs of same-hole pigeons is minimized when you distribute the pigeons as close to evenly as possible subject to this constraint.
Pigeonhole: In the case $r=m+1$, the most even split is $(2,1,1,...,1)$, which has a pair of pigeons in the same hole.