On set-valued cone absolutely summing maps

Spaces of cone absolutely summing maps are generalizations of Bochner spaces L(p)(mu, Y), where (Omega, Sigma, mu) is some measure space, 1 <= p < infinity and Y is a Banach space. The Hiai-Umegaki space L(1)[Sigma, cbf(X)] of integrably bounded functions F : Omega -> cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L(1)(mu, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of L(1)[Sigma, cbf(X)], and to derive necessary and sufficient conditions for a set-valued map to be such a set-valued cone absolutely summing map. We also describe these set-valued cone absolutely summing maps as those that map order-Pettis integrable functions to integrably bounded set-valued functions