凸射影域上距离之间的最优不等式
Optimal inequalities between distances in convex projective domains
摘要：
在实射影空间的任意真凸区域上，存在一个自然的黎曼度量，即Blaschke度量.另一方面，点之间的距离可以用希尔伯特度量来测量。利用最优控制的技巧，我们给出了连接区域上两点的线段的黎曼长度与这两点之间的Hilbert距离的下界不等式，从而加强了Tholozan的一个结果.我们的估计是有效的凸射影域上的黎曼度量，即那些引起凸非退化的中心仿射超曲面浸入的整个类。如果浸入渐近于整环上的凸锥的边界，则我们也可以得到黎曼长度的上界。在这些类上，特别是对于Blaschke度量，我们的不等式是最优的。
英文摘要：
On any proper convex domain in real projective space there exists a natural Riemannian metric, the Blaschke metric. On the other hand, distances between points can be measured in the Hilbert metric. Using techniques of optimal control, we provide inequalities lower bounding the Riemannian length of the line segment joining two points of the domain by the Hilbert distance between these points, thus strengthening a result of Tholozan. Our estimates are valid for a whole class of Riemannian metrics on convex projective domains, namely those induced by convex non-degenerate centro-affine hypersurface immersions. If the immersions are asymptotic to the boundary of the convex cone over the domain, then we can also upper bound the Riemmanian length. On these classes, and in particular for the Blaschke metric, our inequalities are optimal.
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