I saw this argument in the paper a while ago, after reading it I don't really know why BMI continues to be a scale which people (even doctors) use! People who have the same size-to-weight proportions should have the same BMI regardless of their height, right? But they don't - if a short person with a healthy BMI was "grown", so that everything remained in exactly the same proportions, the taller version would have a high BMI.
Here's a simplified example to prove it:
First of all consider a cube with a height of 10cm = 0.1m, and let's say it weighs 1kg. It's BMI is 1/(0.1 x 0.1) = 10
Now imagine you make another cube by placing four of those smaller cubes together - so the new cube has a height of 20cm = 0.2m and a weight of 4 x 1kg = 4kg. It has exactly the same size-to-weight proportions as the smaller cube, it's just twice as big. So it SHOULD have the same BMI, but alas:
4/(0.2 x 0.2) = 100 !!!
The problem with the BMI formula is that it seems to presume people are two dimensional. If you try this same trick ^ with a square you get the same BMIs. So basically if you take a short person and "grow" them into a taller person with the same proportions, the formula allows for all of the extra weight but only two dimensions worth of extra size, instead of three.
Hope you found this as interesting as me!