In many areas of mathematics there are fundamental problems that are embarrasingly natural or simple to state, but whose solution seem so out of reach that they are barely mentioned in the literature even though most practitioners know about them. I'm specifically looking for open problems of the sort that when one first hears of them, the first reaction is to say: that's not known ??!! As examples, I'll mention three problems in geometry that I think fall in this category and I hope that people will pitch in either more problems of this type, or direct me to the literature where these problems are studied.
The first two problems are "holy grails" of systolic geometry---the study of inequalities involving the volume of a Riemannian manifold and the length of its shortest periodic geodesic---, the third problem is one of the Busemann-Petty problems and, to my mind, one of the prettiest open problems in affine convex geometry.
Systolic geometry of simply-connected manifolds. Does there exist a constant $C > 0$ so that for every Riemannian metric $g$ on the three-sphere, the volume of $(S^3,g)$ is bounded below by the cube of the length of its shortest periodic geodesic times the constant $C$?
- For the two-sphere this is a theorem of Croke.
- Another basic test for studying this problem is $S^1 \times S^2$. In this case the fundamental group is non-trivial, but in some sense it is small (i.e., the manifold is not essential in the sense of Gromov).
- There is a very timid hint to this problem in Gromov's Filling Riemannian manifods.
Sharp systolic inequality for real projective space. If a Riemannian metric in projective three-space has the same volume as the canonical metric, but is not isometric to it, does it carry a (non-contractible) periodic geodesic of length smaller than $\pi$?
- For the real projective plane this is Pu's theorem.
- In his Panoramic view of Riemannian geometry, Berger hesitates in conjecturing that this is the case (he says it is not clear that this is the right way to bet).
- In a recent preprint with Florent Balacheff, I studied a parametric version of this problem. The results suggest that the formulation above is the right way to bet.
Isoperimetry of metric balls. For what three-dimensional normed spaces are metric balls solutions of the isoperimetric inequality?
In two dimensions this problem was studied by Radon. There are plenty of norms on the plane for which metric discs are solutions of the isoperimetric problem. For example, the normed plane for which the disc is a regular hexagon.
This is one of the Busemann-Petty problems.
- The volume and area are defined using the Hausdorff $2$ and $3$-dimensional measure.
- I have not seen any partial solution, even of the most modest kind, to this problem.
- Busemann and Petty gave a beautiful elementary interpretation of this problem:
Take a convex body symmetric about the origin and a plane supporting it at some point $x$. Translate the plane to the origin, intersect it with the body, and consider the solid cone formed by this central section and the point $x$. The conjecture is that if the volume of all cones formed in this way is always the same, then the body is an ellipsoid.
Additional problem: I had forgotten another beautiful problem from the paper of Busemann and Petty: Problems on convex bodies, Mathematica Scandinavica 4: 88–94.
Minimality of flats in normed spaces. Given a closed $k$-dimensional polyhedron in an $n$-dimensional normed space with $n > k$, is it true that the area (taken as $k$-dimensional Hausdorff measure) of any facet does not exceed the sum of the areas of the remaining facets?
- When $n = k + 1$ this is a celebrated theorem of Busemann, which convex geometers are more likely to recognize in the following form: the intersection body of a centrally symmetric convex body is convex. A nice proof and a deep extension of this theorem was given by G. Berck in Convexity of Lp-intersection bodies, Adv. Math. 222 (2009), 920-936.
- When $k = 2$ this has "just" been proved by D. Burago and S. Ivanov: http://front.math.ucdavis.edu/1204.1543
- It is not true that totally geodesic submanifolds of a Finsler space (or a length metric space) are minimal for the Hausdorff measure. Berck and I gave a counter-example in What is wrong with the Hausdorff measure in Finsler spaces, Advances in Mathematics, vol. 204, no. 2, pp. 647-663, 2006.