  Recently, a new distance has been introduced for the graphs of two
point-to-set operators, one of which is maximally monotone. When both operators
are the subdifferential of a proper lower semicontinuous convex function, this
distance specializes under modest assumptions to the classical Bregman
distance. We name this new distance the generalized Bregman distance, and we
shed light on it with examples that utilize the other two most natural
representative functions: the Fitzpatrick function and its conjugate. We
provide sufficient conditions for convexity, coercivity, and supercoercivity:
properties that are essential for implementation in proximal point type
algorithms. We establish these results for both the left and right variants of
this new distance. We construct examples closely related to the
Kullback--Leibler divergence, which was previously considered in the context of
Bregman distances, and whose importance in information theory is well known. In
so doing, we demonstrate how to compute a difficult Fitzpatrick conjugate
function, and we discover natural occurrences of the Lambert $\W$ function,
whose importance in optimization is of growing interest.
