Optimal inequalities between distances in convex projective domains

Roland Hildebrand (1)

1 DAO - Données, Apprentissage et Optimisation

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 involving the Hilbert distance between two points of the domain and the Riemannian length of the line segment joining these points, thus strengthening a result of Tholozan. Our estimates are valid for a whole class of complete Riemannian metrics on convex projective domains, namely those induced by centro-affine hypersurface immersions which are asymptotic to the boundary of the convex cone over the domain. On this class our inequalities are optimal.

Mots clés: Hilbert distance; Blaschke metric; centro-affine hypersurface immersion; affine hypersphere

Domaines: Optimisation et contrôle [math.OC]; Géométrie différentielle [math.DG]

Roland Hildebrand. Optimal inequalities between distances in convex projective domains. 2020. ⟨hal-02571624v1⟩