In this paper, fast recursive algorithms for the approximation of an n-dimensional convex polytope by means of an inscribed ellipsoid are presented. These algorithms consider at each step a single inequality describing the polytope and, under mild assumptions, they are guaranteed to converge in a finite number of steps. For their recursive nature, the proposed algorithms are better suited to treat a quite large number of constraints than standard off-line solutions, and have their natural application to problems where the set of constraints is iteratively updated, as on-line estimation problems, nonlinear convex optimization procedures and set membership identification. (C) 2003 Elsevier Ltd. All rights reserved.