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, **e**_{1} and **e**_{2}
as a basis.
Choose the unit ball as

{**e**_{1}*x + **e**_{2}*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

{**e**_{1}*x + **e**_{2}*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

{**e**_{1}*x + **e**_{2}*y | (|x| + |y|) <=
1} is

{**f**_{1}*x + **f**_{2}*y | max(|x|, |y|) <=
sqrt(1/2)} where
**f**_{i}(**e**_{j}) = delta_{ij}.

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.