The aim of this work is to present in a unified approach a series of
results concerning totally convex functions on Banach spaces and their
applications to building iterative algorithms for computing common fixed
points of mea- surable families of operators and optimization methods in
infinite dimen- sional settings. The notion of totally convex function
was first studied by Butnariu, Censor and Reich [31] in the context of
the space lRR because of its usefulness for establishing convergence of
a Bregman projection method for finding common points of infinite
families of closed convex sets. In this finite dimensional environment
total convexity hardly differs from strict convexity. In fact, a
function with closed domain in a finite dimensional Banach space is
totally convex if and only if it is strictly convex. The relevancy of
total convexity as a strengthened form of strict convexity becomes
apparent when the Banach space on which the function is defined is
infinite dimensional. In this case, total convexity is a property
stronger than strict convexity but weaker than locally uniform convexity
(see Section 1.3 below). The study of totally convex functions in
infinite dimensional Banach spaces was started in [33] where it was
shown that they are useful tools for extrapolating properties commonly
known to belong to operators satisfying demanding contractivity
requirements to classes of operators which are not even mildly
nonexpansive.