In this paper centered convex previsions are introduced as a special class of imprecise previsions, showing that they retain or generalise most of the relevant properties of coherent imprecise previsions but are not necessarily positively homogeneous. The broader class of convex imprecise
previsions is also studied and its fundamental properties are demonstrated, introducing in particular a notion of convex natural extension which parallels that of natural extension but has a larger domain of applicability. These concepts appear to have potentially many applications. In this paper they are applied to riskmeasurement, leading to a general definition of convex riskmeasurewhich corresponds, when its domain is a linear space, to the one recently introduced in risk measurement literature.
