
Journal of Convex Analysis 07 (2000), No. 2, 319334 Copyright Heldermann Verlag 2000 Total Convexity for Powers of the Norm in Uniformly Convex Banach Spaces Dan Butnariu Dept. of Mathematics, University of Haifa, 31905 Haifa, Israel Alfredo N. Iusem Inst. de Matématica Pura e Aplicada, Estrada Dońa Castorina 110, Rio de Janeiro, R.J., CEP 22460320, Brazil Elena Resmerita Dept. of Mathematics, University of Haifa, 31905 Haifa, Israel The aim of the paper is to show that, in uniformly convex Banach spaces, the powers of the norm with exponent r > 1 share a property called total convexity. Using this fact we establish a formula for determining Bregman projections on closed hyperplanes and half spaces. This leads to a method for solving linear operator equations (e.g., first order Fredholm and Volterra equations) in spaces which are uniformly convex and smooth. Keywords: Uniformly convex Banach space, totally convex function, duality mapping, Bregman projection. [ Fulltextpdf (686 KB)] 