27 problems

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 $X^*$.

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…

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 $F$ be a Finsler metric on the two-sphere $S^2$, and let a closed geodesic be a periodic geodesic of $F$. Long's elliptic-geodesic conjecture. There exists at least one ellipti…

Hyperbolic closed-geodesic characterization conjecture. The characterization theorem for Finsler metrics, and for Riemannian metrics on closed surfaces, should also hold for revers…

A Finsler manifold $(M,F)$ is homogeneous if its group of isometries acts transitively on $M$. A Finsler metric is Landsberg if its Landsberg curvature vanishes, and it is Berwald…

Let $K\subseteq\mathbb R^d$ be a nondegenerate integral convex polytope containing the origin. A fine structure on $\mathbb R^d$ is required to be $\mathbb Z^d$-periodic, and…

Let $C=C_{a,b}$ be a fine cycle whose two oriented lengths are $a$ and $b$, and let $M$ be a regular fine surface with boundary $C$. Its area is the number of fine triangles it con…

Let $M$ be a surface with a directed Finsler semimetric $F$, filling without shortcuts a Finsler closed curve $(C,G)$. Let $C^+$ and $C^-$ denote the two orientations of $C$, with…

Let $M$ be a surface with a self-reverse Finsler metric, filling isometrically a circle of length $2L$. The Holmes--Thompson area of $M$ is the area of the surface used here, norma…

Let $F$ be a Finsler metric, and let its Landsberg curvature be the corresponding non-Riemannian curvature tensor. A Finsler metric is Berwald when its Berwald parallel transport i…

Let $F$ be a Minkowski norm on $\mathbb{R}^n$, with $n\geq 3$, and let … be its Hessian metric on $\mathbb{R}^n\setminus\{0\}$. Laugwitz's conjecture. If $g$ is flat on…

Let $V_j$ be finite-dimensional real vector spaces equipped with smooth norms $F_j$, and let $e_j\colon M\to V_j$, $j=1,2$, be smooth immersions of a smooth manifold $M$. For a val…

Distance equality conjecture. The distances $d_T$ and $d_F$ coincide on these isometry classes. The inequality $d_T\leq d_F$ is established, and equality is known on Teichmüller sp…

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

Lakzian's conjecture. The space $M$ is diffeomorphic to Euclidean space $\mathbb{R}^{n}$.

Finsler soul conjecture. $M$ is diffeomorphic to

Homogeneous Landsberg–Berwald conjecture. Every homogeneous Landsberg space must be a Berwald space.

Let $F$ be a left invariant $(\alpha_1,\alpha_2)$-metric on a compact connected simple Lie group $G$ with a dimension decomposition $(n_1,n_2)$, where $n_1\geq n_2>1$. Dimension-de…

Let $F$ be a left invariant $(\alpha_1,\alpha_2)$-metric on a compact connected simple Lie group $G$, with a decomposition $\mathfrak{g}=\mathbf{V}_1+\mathbf{V}_2$ such that…

Let $\Sigma$ be a reversible optical hypersurface in the cotangent bundle $T^*\mathbb{R}P^n$ of real projective $n$-space, enclosing a volume $V$. A periodic characteristic on…

Let $(\mathcal{C},d_{\mathcal{C}})$ be a regular Hilbert geometry, and let $\lambda_1(\mathcal{C})$ denote the bottom of the spectrum of its Finsler Laplacian. Rigidity conjecture.…