A Metric Inequality for the Thompson and Hilbert Geometries

Timelike Hilbert and Funk geometries

A timelike space is a Hausdorff topological space equipped with a partial order relation $<$ and a distance function $ρ$ satisfying a collection of axioms including a set of compatibility conditions between the partial order relation and the distance function. The distance function is defined only on a subset of the product of the space with itself that contains the diagonal, namely, $ρ(x,y)$ is defined if and only if $x<y$ or $x=y$. Distances between pairs of distinct points in a triple $x,y,z$, whenever these distances are defined, satisfy the so-called \emph{time inequality}, which is a rev…

A Funk perspective on billiards, projective geometry and Mahler volume

We explore connections furnished by the Funk metric, a relative of the Hilbert metric, between projective geometry, billiards, convex geometry and affine inequalities. We first show that many metric invariants of the Funk metric are invariant under projective transformations as well as projective duality. These include the Holmes-Thompson volume and surface area of convex subsets, and the length spectrum of their boundary, extending results of Holmes-Thompson and Álvarez Paiva on Schäffer's dual girth conjecture. We explore in particular Funk billiards, which generalize hyperbolic billiards in…

具有极小测地线的Finsler圆盘的最小面积

Minimal area of Finsler disks with minimizing geodesics

该研究证明内部测地线均为长度极小的Finsler圆盘的Holmes--Thompson面积下界，构造紧例并发现非旋转对称度量可达等号，通过随机测地线离散化度量证明定理，还得到相关积分几何公式在特定Finsler流形上成立的辅助结果。

Entropy of Hilbert metrics and length spectrum of Hitchin representations in $\mathrm{PSL}(3,\mathbb{R})$

We prove a sharp inequality between the Blaschke and Hilbert distance on a proper convex domain: for any two points $x$ and $y$, \[d^B(x,y) < d^H(x,y) +1.\] We obtain two interesting consequences: the first one is the volume entropy rigidity for Hilbert geometries : for any proper convex domain of $\mathbb{R}\mathbf{P}^n$, the volume of a ball of radius $R$ grows at most like $e^{(n-1)R}$. The second consequence is the following fact: for any Hitchin representation of a surface group into $\mathrm{PSL}(3,\mathbb{R})$, there exists a Fuchsian representation $j$ in $\mathrm{PSL}(2,\mathbb{R})$ s…