On the boundary behaviour of the Riemannian structure of a self-concordant barrier function (Q2730773)

Let \(Q\) be a convex domain in \(\mathbb{R}^n\). A self-concordant function of \(Q\) is a function \(f\) defined on \(Q\), which tends to \(\infty\) at the boundary, the proportions of its derivatives \((f')^2/f''\) and \((f''')^2/ (f'')^3\) are bounded, and for every \(x\) in \(Q\), the Hessian \(g_{ij}(x)= \partial_i \partial_j f(x)\), \(1\leq i\), \(j\leq n\), is positive definite, hence defines a Riemannian structure on \(Q\). The author investigates this Hessian Riemannian structure and its inverse, expresses them by means of partial derivatives of \(e^{-f}\) and the distance from \(x\) to the boundary. He describes the behaviour of the geodesics \(\gamma(t)\) near the boundary, shows its convergence for \(t\to\pm \infty\) to a point \(\gamma(\pm \infty)\in \partial Q\), and this intersection is transversal. He also studies the asymptotic behaviour of the curvature tensor of this Hessian Riemannian structure near the boundary, shows that the sectional curvature converges to \(-1/4\) as \(x\) tends to a suitable part of the boundary. Finally he discusses simple examples of the parabolic domain, the ball, the corner and the triangle.