Journal of Convex Analysis 07 (2000), No. 2, 319--334
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 22460-320, 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.
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