Hilbert度量与Minkowski范数
Hilbert metrics and Minkowski norms
摘要：
证明了有界凸区域$D\subset \mathbb{E}^n$上的Hilbert几何$(D,h_D)$与赋范向量空间$(V,||\cdot ||)$等距当且仅当$D$是开$n$ -单形.关于希尔伯特度量的渐近几何的一个进一步的结果得到了测地线行为的推论。最后证明了Hilbert几何中的每条测地线都收敛于边界上的一点。
英文摘要：
It is shown that the Hilbert geometry $(D,h_D)$ associated to a bounded convex domain $D\subset \mathbb{E}^n$ is isometric to a normed vector space $(V,||\cdot ||)$ if and only if $D$ is an open $n$-simplex. One further result on the asymptotic geometry of Hilbert's metric is obtained with corollaries for the behavior of geodesics. Finally we prove that every geodesic ray in a Hilbert geometry converges to a point of the boundary.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
