球面单形的类时Hilbert几何
Timelike Hilbert geometry of the spherical simplex
摘要：
我们证明了关于单形的类时球面Hilbert几何的以下结果：令$Δ_2$是2-球面上的单形，而$\tildeΔ_2$是对映单形。我们表明，类时球面希尔伯特几何与对$Δ_2, \tildeΔ_2$是等距的工会6个副本的向量空间配备了类时规范，等距和传递的作用后，由组$\mathbb{R}_{>0}^2 \times \mathbb{Z}_3\times \mathbb{Z}_2$。这是一个类时球面模拟的一个众所周知的结果（由于Busemann）指出，希尔伯特度量的欧几里德单纯形是等距的度量诱导的赋范向量空间。同时，这也为类时空间提供了一个新的例子。
英文摘要：
We prove the following result on the timelike spherical Hilbert geometry of simplices: Let $Δ_2$ be a simplex on the 2-sphere and $\tildeΔ_2$ the antipodal simplex. We show that the timelike spherical Hilbert geometry associated with the pair $Δ_2, \tildeΔ_2$ is isometric to a union of six copies of vector spaces equipped with a timelike norm, isometrically and transitively acted upon by the group $\mathbb{R}_{>0}^2 \times \mathbb{Z}_3\times \mathbb{Z}_2$. This is a timelike spherical analogue of a well-known result (due to Busemann) stating that the Hilbert metric of a Euclidean simplex is isometric to a metric induced by a normed vector space. At the same time, this gives a new example of timelike space.
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