Theory of valuations on the space of quasi-concave functions

We studied the space of quasi-concave functions, that are positive real-valued functions defined on R^n with any super-level set either a convex body or empty. We studied mainly the theory of valuations defined on the space of quasi-concave functions, i.e. real-valued functionals with some additivity condition. We established characterization theorems for continuous, with respect to a suitable topology, invariant, with respect several groups that act on R^n, valuations.