After I wrote much of “Isoperimetric, Banach, Area, Crystals” I noticed that the two dimensional analog of my conjecture is false. This leads me to doubt the 3D version. Here are some notes on the 2-D isoperimetric problem.

When we speak of constant area for a convex region there is the problem that 2-D vector spaces come with no defined area concept. It is possible, however, to say that two regions are the same area. The isoperimetric question can thus be reframed: “Of all regions of equivalent area, which has the least perimeter?”. We shall restrict ourselves to convex centrally symmetric regions. It might be wise to declare the area of a unit ball to be 1. I won’t yet.

Choose a pair of independent vectors, e1 and e2 as a basis. Choose the unit ball as
{e1*x + e2*y | (|x| + |y|) <= 1}. (This is the Taxi-cab metric.)

I claim that the minimum perimeter convex set with the area of the unit ball is
{e1*x + e2*y | max(|x|, |y|) <= sqrt(1/2)}. Its perimeter (Sum of Banach lengths) is 4*sqrt(2).

These two convex sets are duals, after some fashion. This duality is suspiciously like the duality between the unit balls for dual Banach spaces. The dual ball for
{e1*x + e2*y | (|x| + |y|) <= 1} is
{f1*x + f2*y | max(|x|, |y|) <= sqrt(1/2)} where
fi(ej) = deltaij.
Recall that vectors in the dual space are linear functions of vectors in the original.

It may lend plausibility to the above to imagine a taxi driver instructed to circle as much area as he can in a city with a regular street grid, but don’t travel more the 20 miles. He is subject to the Taxi-cab metric. He should choose a square route.