On asymptotic volume of Finsler tori, minimal surfaces in normed spaces, and symplectic filling volume
by D. Burago and S. Ivanov

In this ground-breaking paper the authors show that a flat 2-dimensional disc in a finite-dimensional normed space minimizes the Holmes-Thompson area among all other immersed discs with the same boundary. From the convex-geometric viewpoint this is a generalization of Minkowski's theorem on the convexity of the projection body. Indeed, Minkowski's result is equivalent to the statement that hyperplane domains minimize the Holmes-Thompson area. Burago and Ivanov prove their theorem by establishing an equivalence to the volume growth problem for Finsler tori.

Notice that a more general result on the minimality of 2-flats is true and has a simpler proof when the normed space is hypermetric (i.e., when the dual ball is a zonoid). This probably dates back to Busemann, but also follows from the Crofton formulas in hypermetric spaces of Schneider and Wieacker.

Natural variational problems in the Grassmann manifold of a C-star algebra with trace
by C. Durán, L. Mata, and L. Recht

In this paper the authors study a class of (slightly degenerate) Finsler metrics on finite and infinite-dimensional Grassmann manifolds. These metrics are natural from a functional-analytic viewpoint and they have the remarkable property that their geodesics coincide with those of the standard Riemannian metric. The authors show that, despite the degeneracy of these Finsler metrics, the geodesic variational problem can be studied thanks to the particular geometry of the tangent unit spheres. In an appendix, the authors construct non-degenerate Finsler metrics such that their geodesics coincide with those of the standard Riemannian metric. These are all examples of symmetric G-spaces in the sense of Busemann.

Locally Symmetric Finsler Spaces In Negative Curvature
by Patrick Foulon

Abstract. E. Cartan introduced the Riemannian locally symmetric spaces, as the one whose curvature tensor is parallel. They also owe their name to the fact that for each point the geodesic reflection is a local isometry. The aim of this note is to announce a strong rigidity result for Finsler spaces. Namely we show that a negatively curved locally symmetric (in the first above sense) Finsler space is isometric to a Riemann locally symmetric space.

Electronic version.

Analytic continuation of convex bodies and Funks characterization of the sphere
Eric Grinberg and Eric Todd Quinto

Suppose that you have a centrally-symmetric, star-shaped body and that the areas of all intersections of the body with hyperplanes passing through the origin and making a relatively small angle with a given axis are equal. If the boundary is known to osculate a sphere to infinite order along one hyperplane through the axis, then the body is a ball.

The authors also prove a generalization to arbitrary bodies and consider projections instead of intersections. It is particularly interesting to see how microlocal techniques in integral geometry are used to arrive at a result in convex geometry.