Convex Bodies of Constant Width and Constant Brightness
A convex body in three-dimensional Euclidean space is said to be of constant width if the lengths of its orthogonal projections onto lines are all equal. Analogously, the body is said to have constant brightness if the areas of its orthogonal projections onto planes are all equal. A long-standing conjecture was that a convex body of constant width and constant brightness is a sphere. This paper settles the conjecture.
While this result is not directly related to Finsler geometry (yet?). The technique of quasi-conformal maps that is used in the proof should yield new results in Finsler and Minkowski geometry and substitute the complex-analytic methods that are so fruitful in the study of (Riemannian) surfaces.
Remarks on magnetic flows and magnetic billiards, Finsler metrics and a
magnetic analog of Hilbert's fourth problem
The first part of this interesting paper shows that playing magnetic billiards is equivalent to playing billiards in a "Finsler table" where the Finsler metric is conformal to a Randers metric. The second part (section 3) characterizes those Finsler metrics in the plane whose geodesics are circles of a fixed radius. Tabachnikov uncovers an unexpected link to the Pompeiu problem.
Three Papers and a problem on Hilbert Geometries
Hilbert geometries are among the most interesting examples of Finsler metrics. They generalize hyperbolic geometry in various ways: their construction is a natural extension of the Cayley-Klein model of hyperbolic geometry, straight line segments are geodesic, and, when they are regular, their Finsler curvature is constantly -1. On the other hand, as Socié-Méthou showed in her theses, even in the case where the Hilbert metric is a smooth Finsler metric, the Hilbert geometries are not always Gromov hyperbolic.The paper of Benoist Convexes hyperboliques et fonctions quasisymétriques gives a beautiful characterization of those Hilbert geometries that are Gromov hyperbolic.
Karlsson and Foertsch show in Hilbert metrics and Minkowski norms that the only case in which a Hilbert geometry is isometric to a normed space is when the defining convex body is a simplex. On the other hand, Colbois, Vernicos, and Verovic show in L'aire des triangle idéaux en géométrie de Hilbert that, among all the Hilbert geometries, hyperbolic geometry is the only one for which the 2-dimensional Hausdorff measure of all ideal triangles is the same.
Problem. Characterize all Hilbert geometries for which hyperplanes are minimal (extremal) for the Hausdorff measure (or Busemann definition of area).My feeling is that the condition is very strong and that perhaps this will yield a new characterization of hyperbolic geometry.