On Tuesday 06 March 2012 15:43:05 Michael Kleber wrote:
This has nothing to do with justifying its use in shipping, but the function L+W+H on boxes is a pretty important one. ... The key theorem, from the domain of geometric probability, is that the space of continuous invariant measures on polyconvex sets is spanned by only a few basis elements -- in dimension d, they are the d+1 measures that take a box to one of the d+1 elementary symmetric functions of its edge lengths. ... Taking (L,W,H) to L*W*H gives you volume, of course. ... Taking (L,W,H) to 1 gives you Euler characteristic; you should work through the inclusion-exclusion yourself if you haven't seen this before.
There are nice geometric probability interpretations of the measures, too. Suppose A and B are nice convex shapes and A fits inside of B. Then the ratio of volumes is the probability that a randomly-chosen point in B is also in A. The ratio of surface areas is the probability that a randomly-chosen line that passes through B also passes through A, and the ratio of linear girths is the probability that a randomly-chosen plane passing through B also passes through A.
... And, indeed, the ratio of Euler characteristics is the probability that a randomly-chosen all-of-R^n passing through B also passes through A, since any convex set[1] has Euler characteristic 1. :-) [1] Er, I think. Certainly any compact convex set, which I think is the correct condition for the conditional-probability theorem to apply. -- g