  In the present contribution we characterize law determined convex risk
measures that have convex level sets at the level of distributions. By relaxing
the assumptions in Weber (2006), we show that these risk measures can be
identified with a class of generalized shortfall risk measures. As a direct
consequence, we are able to extend the results in Ziegel (2014) and Bellini and
Bignozzi (2014) on convex elicitable risk measures and confirm that expectiles
are the only elicitable coherent risk measures. Further, we provide a simple
characterization of robustness for convex risk measures in terms of a weak
notion of mixture continuity.
