5 problems

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

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.…

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…

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