  In this paper we study two classes of imprecise previsions, which we termed
convex and centered convex previsions, in the framework of Walley's theory of
imprecise previsions. We show that convex previsions are related with a concept
of convex natural estension, which is useful in correcting a large class of
inconsistent imprecise probability assessments. This class is characterised by
a condition of avoiding unbounded sure loss. Convexity further provides a
conceptual framework for some uncertainty models and devices, like unnormalised
supremum preserving functions. Centered convex previsions are intermediate
between coherent previsions and previsions avoiding sure loss, and their not
requiring positive homogeneity is a relevant feature for potential
applications. Finally, we show how these concepts can be applied in (financial)
risk measurement.
