Exact capacities and star shaped distorted probabilities.

We are interested in this work, in capacities which are deformations of probability i.e. v = f o P, we characterize respectively balanced, totally balanced, exact and convex capacities by properties concerning the probability transformation function f. And we give the explicit expression, in the case of a convex capacity v = f o P, of a probability in the core of v which coincides with v on a given finite chain of elements of the algebra A. We end this work by two notions of increase in risk recently studied in [4] and connected with star-shaped functions and with an application to the optimality of the deductible insurance studied in [14].