Linearity with Multiple Priors

We characterize the types of functions over which the functional defined as the "min" of integrals with respect to probabilities in a given non-empty closed and convex class is linear. This happens exactly when "integrating" functions which are positive affine transformations of each other (or when one is constant). We show that the result is quite general by restricting the types of classes of probabilities considered. Finally we prove that, with a very peculiar exception, all the results hold more generally for functionals which are linear combinations of the "min" and the "max" functional.