4
votes
0answers
223 views

Symmetric matrices and Hilbert's fourth problem

From the analytic viewpoint, the Busemann-Pogorelov solution of Hilbert's fourth problem is summarized in the following result: Theorem. All straight lines are extremals of the variational problem $$ ...
2
votes
1answer
263 views

Reversibility vs geodesic reversibility for Finsler metrics on the two-sphere

Problem. To give a concrete example of a geodesically reversible Finsler metric on the two-sphere that is not just the sum of a reversible Finsler metric and an exact $1$-form. Some background may be ...
5
votes
1answer
196 views

Forms satisfying the zero-energy condition on the projective plane

Theorem (Michel). A $1$-form on the projective plane is exact if and only if its integral over any projective line is equal to zero. Is there a simple proof of this result due, I think, to R. Michel ...
3
votes
1answer
368 views

On the generalized Radon transform and currents

Given a family of hypersurfaces $H_{t,p} = $ {$x \in \mathbb{R}^n \mid g(x,p) = t $} one defines a generalized Radon transform $R$ of a function $u \colon \mathbb{R}^n \to \mathbb{R}$ as $$ R[u] ...