  The Bregman divergence (Bregman distance, Bregman measure of distance) is a
certain useful substitute for a distance, obtained from a well-chosen function
(the "Bregman function"). Bregman functions and divergences have been
extensively investigated during the last decades and have found applications in
optimization, operations research, information theory, nonlinear analysis,
machine learning and more. This paper re-examines various aspects related to
the theory of Bregman functions and divergences. In particular, it presents
many sufficient conditions which allow the construction of Bregman functions in
a general setting and introduces new Bregman functions (such as a negative
iterated log entropy). Moreover, it sheds new light on several known Bregman
functions such as quadratic entropies, the negative Havrda-Charv\'at-Tsallis
entropy, and the negative Boltzmann-Gibbs-Shannon entropy, and it shows that
the negative Burg entropy, which is not a Bregman function according to the
classical theory but nevertheless is known to have "Bregmanian properties",
can, by our re-examination of the theory, be considered as a Bregman function.
Our analysis yields several by-products of independent interest such as the
introduction of the concept of relative uniform convexity (a certain
generalization of uniform convexity), new properties of uniformly and strongly
convex functions, and results in Banach space theory.
