The reach property for self-concordant barriers

Let K be a regular convex cone in R^n. A smooth locally strongly convex logarithmically homogeneous function on the interior of the cone is called a self-concordant barrier if it tends to +infty on the boundary of the cone and satisfies a certain inequality between the second and third derivatives. Self-concordant barriers can be equivalently described by hyperbolic centro-affine immersions which are asymptotic to the boundary of the cone and have a bounded cubic form. Hyperbolic centro-affine hypersurface immersions in R^n can in turn be seen as positive definite Lagrangian submanifolds of a certain para-Kähler space form. The cubic form of the immersion becomes the second fundamental form of the submanifold. We show that Lagrangian submanifolds which arise in this way have the global reach property, with parameter given by a function of the bound of the cubic form.