Central path and Riemannian distances Yu. Nesterov * and A. Nemirovski † June, 2003 Abstract In this paper we study the Riemannian length of the primal central path computed with respect to the local metric defined by a self-concordant function. We show that despite to some examples, in many important situations the length of this path is quite close to the length of geodesic curves. We show that in the case when the Riemannian structure of a bounded convex set is introduced by a ν -self-concordant barrier, the central path is sub-geodesic up to the factor ν 1/4 . Keywords: Riemannan geometry, convex optimization, structural optimization, interior-point meth- ods, path-following methods, self-concordant functions, polynomial-time methods. * CORE, Catholic University of Louvain, 34 voie du Roman Pays, 1348 Louvain-la-Neuve, Belgium; e-mail: nesterov@core.ucl.ac.be. This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming. † Technion-Israel institute of Technology, Haifa, Israel; e-mail: nemirovs@ie.technion.ac.il