  This paper introduces and fully characterizes the novel class of
quasi-logconvex measures of risk, to stand on equal footing with the rich class
of quasi-convex measures of risk. Quasi-logconvex risk measures naturally
generalize logconvex return risk measures, just like quasi-convex risk measures
generalize convex monetary risk measures. We establish their dual
representation and analyze their taxonomy in a few (sub)classification results.
Furthermore, we characterize quasi-logconvex risk measures in terms of
properties of families of acceptance sets and provide their law-invariant
representation. Examples and applications to portfolio choice and capital
allocation are also discussed.
