There is a much wider generalization here which is studied under the name of "property elicitation" in computer science, machine learning, and statistics.

The generic question is: Given a loss function, what "property" of the distribution minimizes average loss; and given a "property", characterize all such loss functions. For example, Bregman divergences are (essentially) all losses that "elicit" the mean of a distribution. If you have any monotone continuous function g(), then |g(x) - g(s)| actually also elicits the median, and these are essentially all that do.

Apologies for self-promotion, but you can read more at references on this page (disclaimer: I'm one of the researchers who posted it): https://sites.google.com/site/informationelicitation/

or tutorials on this subject at my blog: http://bowaggoner.com/blog/series.html#convexity-elicitation