Dimensional Dilemma

Just a short post, since I got an apartment earlier this week, and there's still lots of set up to do!

While planning for this move to France, I had to figure out how to fit everything I want to have the next 2 years into airline-approved bags, and I found their size limit for checked luggage interesting. I would have assumed the limit would be volume:

where x, y, and z are the length, width, and height. The trouble with this though, is that it would allow narrow objects with enormous length, like a long pole (and you don't want to get a physicist started on fitting poles into things). Instead, airlines place a limit on the sum of the dimensions:

What's interesting is that this also places a limit on the volume of the object. We get the maximum volume when x = y = z, so

It seems like we're getting extra information here: With only one expression we can limit both the volume and the maximum size. It's not entirely clear to me where this information comes from, but my guess is that it's due to the fact that the linear limit defines a coordinate system, in this case Cartesian. We could imagine a different limit, where items were required to fit inside a sphere of radius r. This would also limit the volume:

Depending on how you choose the linear measurement, you can get different maximum volumes. Makes me wonder whether airlines initially used the volume limit, only to be inundated with pole-vaulters...