90 problems

Busemann's conjecture. Every G-space is finite-dimensional, and every finite-dimensional G-space is a topological manifold.

A prime closed geodesic on a Finsler manifold is a closed geodesic that is not an iteration of another closed geodesic. For a Finsler metric on the sphere $S^n$, let $[x]$ denote t…

A Finsler metric $F$ on the sphere $S^d$ is an irreversible Finsler metric defining the geodesic flow under consideration, and distinct closed geodesics are counted as geometricall…

Dušek's conjecture. Every homogeneous Finsler manifold admits at least two homogeneous geodesics through every point.

Global Holmes–Thompson isoperimetric conjecture. The circle $c_0$ encloses the maximal Holmes–Thompson area among all such curves.

Let $L$ be a reversible Finsler metric on ${\mathbb R}P^2$. A smooth positive function $\rho$ on ${\mathbb R}P^2$ is a function whose product with $L$ defines the conformally resca…

Schäffer's girth duality conjecture. The girth of $X$ equals the girth of $X^*$.

Shen's conjecture. There are non-trivial positively complete, projectively flat Finsler metrics of constant curvature

Let $u$ be a solution of the Monge–Ampère equation in a setting where the associated function $u^2$ may fail to be smooth, and let $g$ denote the Riemannian metric used in the smoo…

Let $F$ be a non-Riemannian almost rational Finsler metric (AR-Finsler metric) on a smooth manifold $M$. A Finsler metric is locally dually flat if, in suitable local coordinates,…

Let $F$ be a Finsler metric on $S^2$, and call a closed geodesic prime if it is not an iterate of a shorter closed geodesic. Two-or-infinity conjecture for Finsler metrics. Every F…

Let $(M,S,F^{\pm})$ be a bipartite space, where $S$ is the slit, $S_x=\ker(s_x)$ is its fibre at $x\in M$, and $F^-$ is the negative partial Finsler function. The apple bipartite-s…

Let $(M,S,F^{\pm})$ be a bipartite space, where $S$ is the slit and $F^+$ is its positive partial Finsler function. The lemon bipartite-space conjecture. A bipartite space of the f…

Reid's conjecture. $$ -Finsler metrics are determined by their length spectra, and these length spectra form a dense subset of

Consider a Finslerian spacetime geometry whose Lagrangian depends on direction and whose gravitational field includes the contribution of the velocity distribution of a many-partic…

Let $M$ be spacetime, let $L$ be a Finsler Lagrangian, and let $\mathcal{S}_x$ denote the set of unit timelike directions at $x\in M$. For a 1PDF $\varphi(x,\dot x)$, the second mo…

Let $W$ be the distance map associated with a data-driven Finslerian car model on a Finsler manifold, let $\mathcal{F}^*$ be the dual Finsler function, and let $\mathbf{p}_0$ be th…

Let $\varphi(x,\dot x)$ be the one-particle distribution function (1PDF) of a cosmological kinetic gas on the tangent bundle of spacetime, and let … be its moments, where…

Let $g_1,\dots,g_k$ be a family of positively related Riemannian metrics on a homogeneous space. A family is positively related when its metrics admit a common decomposition … with…

Let $k$ be a positive integer, and let $S^{2k}$ or $S^{2k-1}$ be an even- or odd-dimensional sphere equipped with a Finsler metric. Katok–Anosov conjecture. Every Finsler metric on…

The Teichmüller metric and the $L^1$-norm on holomorphic quadratic differentials form an example of dual Finsler structures, related through reflexive duality. Reflexive duality su…

Let $\Sigma$ be a closed surface that is not a torus or a Klein bottle, and let $(K_x)_{x\in\Sigma}$ be a continuous field of convex bodies with $K_x\subset T_x\Sigma$. Suppose tha…

Sharp $4/\pi$ comparison conjecture. The inequalities

Triangular Finsler boundary conjecture. The quotient $\Delta Fins(S)/\mathbb{R}_+$ injects into $\mathbb{P}(\mathbb{R}^{\mathcal{L}(S)})$ and comprises an open dense subset of the…

Let $(M,F)$ be a compact simply connected $n$-dimensional irreversible Finsler manifold. A prime closed geodesic is a closed geodesic that is not an iteration of another closed geo…