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 develop inequalities that provide lower bounds for the Riemannian length of the line segment joining two points of the domain by the Hilbert distance between these points. This strengthens 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.

在实际投影空间中的任何合适的凸域上，都存在一个自然的黎曼度量，即Blaschke度量。另一方面，可以使用希尔伯特度量标准来测量点之间的距离。使用最佳控制技术，我们开发了不等式，该不等式为通过两个点之间的希尔伯特距离将连接该域的两个点的线段的黎曼长度提供了下限。这加强了Tholozan的结果。我们的估计对于凸射影域上的一整类黎曼度量是有效的，也就是由凸非简并中心仿射超表面浸没引起的。如果浸入对于该区域上凸锥的边界是渐近的，那么我们也可以将Riemmanian长度上限。在这些类上，尤其是对于Blaschke度量，我们的不等式是最优的。