This has nothing to do with justifying its use in shipping, but the function L+W+H on boxes is a pretty important one. I thought it was called something like "linear girth", but I can't find any justification for that online, so seems likely that my memory is faulty. But I'll use that here since I need some name for it. 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. You extend the measures by additivity on disjoint sets, and so by mu(A U B) = mu(A) + mu(B) - mu(A ^ B) on arbitrary sets. (Using ^ for intersection here.) Taking (L,W,H) to L*W*H gives you volume, of course. Taking (L,W,H) to LW+WH+HL gives you surface area (up to a scalar), and here you need to think about the mu(A ^ B) term: a unit cube gets mu=3, while a 1x1x2 block gets mu=5 because you can break it into two unit cubes that intersect on a 1x1x0 square with valuation 1. Taking (L,W,H) to L+W+H gives you the linear girth thing under discussion here. 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. --Michael On Mon, Mar 5, 2012 at 7:43 PM, Henry Baker <hbaker1@pipeline.com> wrote:
Very interesting! Thanks!
I wonder if "length" is treated differently because of a preferred direction ("This Side Up").
Or perhaps ships & airplanes are long & thin, so "length" is treated differently for those reasons.
Although, the airplane shipping containers that I have seen look like 1/2 of a sliced cylinder -- probably because they are loaded below the "deck". These airline shipping containers are also not very long -- less than 6' -- so their length is less than their radius.
I'll have to do some research to see how far back L*G goes in the shipping biz.
So for 2 dimensions we have "perimeter" and "girth" (presumably the perimeter of the convex hull); there doesn't seem to be a name for L+W+H in 3 dimensions. Perhaps there is a name for supremum girth, over all 2-D projections of the object.
At 04:12 PM 3/5/2012, James Cloos wrote:
> "HB" == Henry Baker <hbaker1@pipeline.com> writes:
HB> So, given these constraints, is f(L,W,H)=C*(L+W+H) even close to optimal?
Probably not.
The package shipping companies (USP, USPS, FedEx, et al) tend to use lenght * girth, making that f(L,W,H) = C * L*(W+H). I've heard -- but cannot confirm -- that L*G traces back to nautical shipping.
I can confirm that USP and USPS, at least, were already using L*G back in the '70s.
-JimC -- James Cloos <cloos@jhcloos.com> OpenPGP: 1024D/ED7DAEA6
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