On the boundary behaviour or the Riemannian structure of a self-concordant barrier function

Abstract

Following Vinberg [18], we will dene a convex domain in R n as an open convex subset of R n which does not contain a full straight line. A self-concordant barrier function for a convex domain Q is dened as a strongly convex smooth function f on Q, which tends to 1 at the boundary and satises the estimates (iii) and (iv) of Denition 2.1 below, for the derivatives of f up to the third order. The strong convexity off means that the Hessian gij (x) =@i@j f(x) is positive denite and therefore denes a Riemannian structure on Q. (Such Riemannian structures on convex domains have been studied already by Koszul [13] and Vinberg [18], who refer further back to the theory of bounded domains in C n , with its Bergmann metric.) In this paper we investigate the asymptotic behaviour of this Riemannian structure, and of its geodesics and its curvature, near points of the boundary where the boundary is smooth and strongly convex, which means that its curvature, described by its second fundamental form, is positive denite.