a [slightly advanced] mathematical puzzle

[mehtank]

to keep your brain un-dead over the holiday. actually, the more i think about it, the advanced-er it becomes...

the thought came to me with the talk of possible objects that's been going around: what happens if you overinflate an inflatable cube? it seems to be an interesting mathematical problem, so i'll phrase it [a bit] more rigorously.

what is the maximum volume of a region completely bounded by 6 smooth surfaces, subject to the following conditions:
~ each surface is of unit area, bounded by 4 continuous curves of unit length intersecting at right angles
~ two surfaces intersect completely along a common edge, and are orthogonal at every point along their intersection
~ three surfaces come together at a corner

i think those are the only conditions... basically you have a unit cube made of rubber walls, but where the edges and vertices maintain fixed relations. what shape will give it the maximum enclosed volume, and what is that volume?