I've only played with this a little for the past day or so, and haven't thought about it too hard, so it might be obvious. Obviously it's not fair to ask for a "combinatorial proof" of an inequality involving real numbers, so we'll ask that the vectors be in $\mathbb{N}^n$. More concretely:

Given n boxes subdivided into a "right half" and a "left half" with $a_i$ objects in the right half of box $i$, and $b_i$ in the left half of box i, is there a natural injective function from

Two pairs (ordered, with replacement) of objects, with each pair containing one object from the left half and one object from the right half of a fixed box

to

A pair (ordered, with replacement) from the right half of some box, and a pair (O,WR) from the left half of some (possibly different) box?

(Sorry if this is a double; my wireless is being strange.)

The Cauchy-Schwarz Master Classby J. Michael Steele. $\endgroup$ – Ian Durham Feb 23 '10 at 12:30