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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 $$ …

The following nonlinear integral transform takes continuous functions defined on the cylinder $\mathbb{R} \times S^1$ to $C^2$ functions defined on the plane $\mathbb{R}^2$: $$ \mathcal{A}f (x,y) := \ …

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 …

The following text is an attempt to see Hilbert's fourth problem in a new light. Definition. A pseudometric $d$ on $\mathbb{R}^n$ is called projective if whenever a point $z$ belongs to a line segme …

Given a nowhere-zero, closed $2n$-form $\Omega$ in a manifold of dimension $2n +1$, how do we know if there exists a closed $2$-form $\omega$ such that $\Omega = \omega^n$? Remark. This question star …

A classical result is that solutions of the isoperimetric problem on the plane, the sphere, and the hyperbolic plane are circles. Are there any other Riemannian metrics on these spaces that share this …