Problems worthy of attack prove their worth by hitting back — Piet Hein
In all the problems that follows, we shall use the word volume or area to refer to the Holmes-Thompson volume (or area functional). Hausdorff measure and Gromov's mass* will be simply called by their proper names.
The geometry of normed spaces
1. Given a non-Euclidean three-dimensional normed space, is it possible to find on its unit sphere a closed curve that is symmetric with respect to the origin and whose length is strictly less than $2\pi$?
Comment by J.-C Alvarez Paiva. This question was posed by J.-J. Schäffer in his book Geometry of spheres in normed spaces. In a conversation we had in 2002, Schäffer mentioned that he thinks such a curve always exists and that it may be taken to be planar.
2. Give a sharp lower bound for the two-dimensional Hausdorff measure of unit spheres on three-dimensional normed spaces.
Comment by J.-C Alvarez Paiva. The best lower bound so far is $36/\pi$ and is due to Alvarez-Paiva, Ivanov, and Thompson. Moreover, this bound is sharp for the Holmes-Thompson area.
3. Given a two-dimensional closed polyhedral surface in a four-dimensional normed space, is the area of any given face not more than the sum of the areas of the remaining faces?
Comment by J.-C Alvarez Paiva. When the surface is a poyhedral two-sphere this was proved by D. Burago and S. Ivanov in On asymptotic volume of Finsler tori, minimal surfaces in normed spaces, and symplectic filling volume (Annals of Math., 2002). If we replace area by two-dimensional Hausdorff measure the problem has been open for over sixty years (even in the case of polyhedral spheres).
4. In a four-dimensional normed space $X$ all hyperplanes are isometric. Is $X$ necessarily Euclidean?
Comment by J.-C Alvarez Paiva. This is a particular case of a question of Banach and this is the easiest case not solved by Gromov in On a geometric hypothesis of Banach (1967). I think Finsler geometry may be useful in solving this problem.
Funk metrics, Thompson metrics, and Hilbert geometries
Hilbert's fourth problem and inverse problems
1. Give a non-trivial example of a reversible Finsler metric on the complex projective plane for which (unparameterized) geodesics coincide with those of the standard Riemannian metric.
Comment by J.-C Alvarez Paiva. This is Problem 11 in my Problems on Finsler geometry. I still have no idea on how to find such a metric or on whether it even exists. The general aim is to extend Hilbert's fourth problem to all rank-one symmetric spaces.
Geodesic flow and dynamics
1. Find a reversible Finsler metric on the two-sphere whose geodesic flow is not symplectically conjugate to the geodesic flow of a Riemannian metric.
Comment by J.-C Alvarez Paiva. This is Problem 9 in my Problems on Finsler geometry. Note that the geodesic flow of the Katok examples of non-reversible Finsler metrics on the two-sphere with only two closed prime geodesics cannot be conjugate, or even orbitally equivalent, to geodesic flows of Riemannian metrics.
Systolic geometry
1. The Hausdorff measure of a reversible Finsler two-torus $(T^2,F)$ equals $\pi/4$. Is there a non-contractible closed curve in $(T^2,F)$ of length at most $1$?
Comment by J.-C Alvarez Paiva. When the torus is flat this is Minkowski's lattice point theorem in two dimensions. In the general case S. Saborau has given the following rougher estimate: if the Hausdorff measure of a reversible Finsler two-torus $(T^2,F)$ equals $2/\pi$, it contains a non-contractible closed curve of length at most $1$.
2. The area of a non-reversible Finsler metric $(RP^2,F)$ equals $2\pi$. Is there a non-contractible closed curve in $(RP^2,F)$ of length at most $\pi$?
Comment by J.-C Alvarez Paiva. Ivanov has given an affirmative answer when the Finsler metric is reversible. The results of Alvarez-Paiva and Balacheff suggest
that the answer will still be affirmative for non-reversible metrics.
Connections and curvature
Minimal surfaces and variational calculus
1. Does every three-dimensional Minkowski space admit ruled minimal surfaces other than planes?
Comment by J.-C Alvarez Paiva. I haven't had the time to really think about this problem myself. The general aim is to construct many explicit (and geometric) examples of minimal surfaces in Finsler spaces that are not also totally geodesic. Recall that G. Berck has proved that totally geodesic submanifolds are minimal (Math. Ann., 2009), but these are about the only class of examples of minimal submanifolds in our possession.
