
Journal of Convex Analysis 10 (2003), No. 1, 035061 Copyright Heldermann Verlag 2003 On Uniform Convexity, Total Convexity and Convergence of the Proximal Point and Outer Bregman Projection Algorithm in Banach Spaces Dan Butnariu Dept. of Mathematics, University of Haifa, 31905 Haifa, Israel, dbutnaru@math.haifa.ac.il Alfredo N. Iusem Institute of Pure and Applied Mathematics (IMPA), Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ, CEP 22460320, Brazil, iusp@impa.br Constantin Zalinescu Faculty of Mathematics, "Al. I. Cuza" University, 6600 Iasi, Romania, zalinesc@uaic.ro We study and compare the notions of uniform convexity of functions at a point and on bounded sets with the notions of total convexity at a point and sequential consistency of functions, respectively. We establish connections between these concepts of strict convexity in infinite dimensional settings and use the connections in order to obtain improved convergence results concerning the outer Bregman projection algorithm for solving convex feasibility problems and the generalized proximal point algorithm for optimization in Banach spaces. Keywords: Uniform convexity at a point, total convexity at a point, uniform convexity on bounded sets, sequential consistency, generalized proximal point algorithm for optimization, outer Bregman projection algorithm for feasibily. MSC 2000: 90C25, 90C48, 26B25, 49J52, 46N10, 46N20, 90C30. FullTextpdf (629 K) 