Total convexity for powers of the norm in uniformly convex Banach spaces

We show that in any uniformly convex Banach space the functions f(x) = \ \x \ \ (r) with r is an element of (1, infinity) are totally convex. Using th is fact we establish a formula for determining Bregman projections on close d hyperplanes and half spaces. This leads to a method for solving linear op erator equations (e.g., first kind Fredholm and Volterra equations) in spac es which are uniformly convex and smooth.