  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.
