Isoperimetric, Banach, Area, Crystals

I have found no references to concepts for area of embedded sub-manifolds except for locally Euclidean spaces. I want to suggest that area of surfaces in Banach spaces is an easy and perhaps useful generalization. I restrict myself here to finite dimensional vector spaces over the reals. A norm on a vector space is a function from vectors to reals. ||x|| is a real when x is a vector and:

for x a vector and c a real: ||cx|| = |c| · ||x||
||x+y|| ≤ ||x||+||y||
||x|| ≥ 0
if ||x|| = 0 then x = 0

This provides a metric: d(x,y) = ||x−y|| and thus a topology. The set of points x such that ||x||<1 is a centrally symmetric convex set sometimes called the unit ball which determines the norm.

We will use letters u and v to name vectors from the unit ball.
||u|| ≤1, ||v|| ≤ 1.

A norm may substitute for an inner product in many ways. If (,) is an inner product then ||x|| = sqrt((x,x)) is a norm.

The set of real valued linear functions on a vector space is itself a vector space which is called the dual space of the original space. We define a norm in the dual space thus:

||f|| = max_uf(u).

This definition defines a dual norm in the dual space of a normed space. The dual dual norm is the original norm.

The above is conventional Banach theory. The quality of what follows is substandard mathematically. Perhaps someone will prove, improve or disprove some of the conjectures. It will also be necessary to improve definitions. (I currently think that the crystal conjecture is wrong, but I think it can be repaired!)

Area

The Greeks explained the quantitative concept of area by assigning an area to small squares, dividing larger areas into these small squares and summing the areas. To follow the Greeks we must assign an area to small parallelograms, there being no concept of right angle available. Exterior differential forms, or just exterior products, provide a quantitative vector magnitude for these. We need a scalar area concept, however.

The oriented area of a parallelogram may be taken to be a vector in the space A of alternating real valued bilinear functions over the dual of the space in which the surface is embedded. If we had a norm for A we could assign an area to our small parallelograms.

For the real valued bilinear function f(x, y) I propose the norm ||f|| = max_uv f(u, v). A Banach style area can now be defined as the integral of norm of small area elements.

I conjecture that if we take a crystal centered at the origin as the unit ball of a Banach space, then the shape for constant volume that minimizes the Banach style area is similar and parallel to the crystal itself. Thus as an isotropic water droplet assumes a spherical shape to minimize its Euclidean surface, then so does a crystal assume its shape to minimize its Banach style surface area. In other words a crystal assumes the shape of a Banach sphere.

Afterthoughts

Stefan Banach contributed to the problem of the definition of area in Euclidean space and so “Banach Area” may be a confusing term.

When f(u,v) takes on its maximum value u and v are sort of orthogonal. (the unit ball is compact; f achieves its supremum.)

This generalizes to areas (content) of p-branes within n-spaces.

If a unit ball is a polytope then so is its dual unit ball.

A less elegant possibility is that the minimized area (energy) is based on a norm for the space of oriented areas and that this norm is not produced by any norm on the space of the crystal.

π = 3 for the norm ||<x, y>|| = max(|x|, |y|, |x+y|)
π = 4 for the norm ||<x, y>|| = max(|x|, |y|)
Another conjecture: 3≤ π ≤4.

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