When the sphere is not round I.

This is the first of a series of posts concerning the geometry of unit spheres in normed spaces with the aim of attracting the attention of fellow geometers to a number of delightful open problems.  I will start with a couple of problems posed by Juan Jorge Schäffer in his book ”Geometry of Spheres in Normed Spaces”, Lecture Notes in Pure and Appl. Math., vol. 20, Dekker, NY, 1976.

Let us define the girth of a normed space (X,\|\cdot\|) as the infimum of the lengths of all rectifiable, centrally-symmetric, closed curves on the unit sphere. This quantity can also be described in (at least) two other ways : (1) as twice the infimum of  the (intrinsic) distance between a pair of antipodal points on the unit sphere; (2) as twice the infimum of the lengths of non-contractible rectifiable curves on the projectivized space P(X) with the length metric induced from the unit sphere. This second description links the girth to the study of systoles in projective spaces endowed with Finsler metrics.

In his book, Schäffer made the following two conjectures about the girth of normed spaces—he does not assume the spaces are Banach:

Conjecture 1. The girth of a normed space is equal to the girth of its dual.

Conjecture 2. The girth of a normed space of dimension greater than two is at most 2\pi with equality if and only if the space is Euclidean.

Some years ago I was lucky enough to settle the first conjecture in the affirmative by using some  elementary considerations in symplectic geometry (the paper was published in the American Journal of Mathematics, 2006, vol. 128, no2, pp. 361-371, but you can also find it in the ArXiv). Therefore, it is really the second conjecture that I would like to talk about.

To understand why the conjecture is plausible we must recall a little gem of the theory of normed spaces: suppose you are a being living in a normed space of dimension two and you’ve heard somewhere that one of the most important constants in the universe is half the perimeter of the unit disc, so you go and measure it. What do you get? A Polish mathematician by the name of Golab did just that and he found that the value of “\pi” on a normed plane is between 3 and 4. Moreover, you only get three if your unit disc is linearly equivalent to a regular hexagon and you only get four if  your unit disc is a parallelogram. Notice furthermore that the girth of a normed plane is exactly the perimeter of its unit disc, so–in that case—having girth greater than 2\pi means ”squarish” (OK, this is not rigourous, but perhaps making it so is not worthwhile at this point).  Now take the unit ball of a normed space and consider its two-dimensional central sections (i.e., its intersections with two-dimensional subspaces). The perimeters of the unit discs in these sections are greater than or equal to the girth of the space. Therefore if the girth is greater than 2\pi, all the sections are “squarish”.

A vaguely stated principle in convex geometry is that of all the central sections of a symmetric convex body some are rounder (i.e. more elliptical) than others. In my view we (or at least I) are far  from understanding this principle, but it does find its way into some results such as

Dvoretzki’s theorem: Given a positive number \epsilon and a positive integer k, there exists a positive integer N(\epsilon,k) such that any normed space of dimension greater than N(\epsilon,k) has a k-dimensional subspace that is \epsilon-close—in the Banach-Mazur distance—to a Euclidean space.

In particular, the girth of an infinite-dimensional normed space is at most 2\pi. Schäffer remarks in his book that it is enough to prove the conjecture for three-dimensional normed spaces.

What puzzles me in approaching Schäffer’s conjecture is that I’m not sure whether the key to its solution is geometric or topological. When the sun is out and I feel optimistic, I start believing that the following question has an affirmative answer.

Question. Let \pi : E \rightarrow S^2 be a non-trivial two-dimensional vector bundle over the sphere. Assume that on every fiber E_x we have a norm \|\cdot\|_x varying continuously with the base point. Is it true that at some point z \in S^2 the perimeter of the unit disc in E_z is less than or equal to 2\pi ?


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