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Total convexity for powers of the norm in uniformly convex Banach spaces. (English) Zbl 0971.46005
Summary: 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 Bergman 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.

##### MSC:
 46B20 Geometry and structure of normed linear spaces 47A50 Equations and inequalities involving linear operators, with vector unknowns
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