15 problems

Let $D$ be a bounded convex domain in a finite-dimensional real vector space, endowed with its Hilbert metric $d_H$, and let $f:D\rightarrow D$ be a fixed-point-free nonexpansive m…

Let $K\subset\mathbb{R}^n$ be a convex body, and equip its interior with the Hilbert metric. The volume growth entropy measures the exponential growth rate of metric balls in this…

Let $(Y,F_\Omega)$ be a compact strictly convex real projective manifold of dimension at least $3$, and let $(X,g_0)$ be a hyperbolic structure on the same underlying manifold. Let…

Let $D$ be a cross-section of a proper open cone, and let $\operatorname{Isom}(D)$ and $\operatorname{Coll}(D)$ denote respectively the isometry group and the collineation group of…

Karlsson–Nussbaum conjecture. There exists a convex set $\Lambda$ in $\partial\Omega$ such that

Bletz-Siebert–Foertsch conjecture. The Euclidean rank of any Hilbert geometry is $1$.

Lemmens–Walsh conjecture.

Let $C$ be a bounded convex domain in $\mathbb{R\mathrm{^{N}}}$ equipped with Hilbert's metric $d$, and let $f:C\rightarrow C$ be a 1-Lipschitz map. Nussbaum–Vandehey conjecture. E…

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.…

Asymptotic volume conjecture.

Let $\Omega\subset\mathbb{R}\mathbb{P}^n$ be a convex proper open set, and let $h_{vol}(\Omega,d_{\Omega})$ denote the volume entropy of its Hilbert geometry, defined using the Bus…

Symmetric-cone conjecture. The groups $\mathrm{Isom}(X)$ and $\mathrm{Coll}(X)$ differ if and only if the cone generated by $X$ is symmetric and not Lorentzian. In that case, the i…

de la Harpe's conjecture. The group $\mathrm{Isom}(X)$ is a Lie group, and $\mathrm{Isom}(X)$ acts transitively on $X$ if and only if $\mathrm{Coll}(X)$ acts transitively on $X$.

Let $(\Omega,d_{\Omega})$ be a Hilbert geometry, where $\Omega$ is a convex set of dimension $n$. If the volume entropy exists, define it by … Here $B(x,r)$ is a Hilbert-metric bal…

Amenability characterization. A Hilbert geometry is amenable if and only if it is a $G_n$-polygon.