Convergence of asymptotic directions (Q2723466)

In this interesting paper the authors study convergence properties of asymptotic cones in relation with continuity properties of set-valued maps. Beside the usual lower, upper and upper Hausdorff continuity, closedness and boundedly compactness for set-valued maps, they use adequate notions for cone-valued maps like cosmic upper and lower continuity, cosmic closedness and conic upper continuity. Results relating these notions, as well as properties for the union, intersection, Cartesian product and composition of cone-valued maps are established. Associating to the set-valued map \(M:\Omega \rightarrow 2^{X}\) the cone-valued map \(R_{M}:\Omega \rightarrow 2^{X}\) defined by \(R_{M}(\omega)= \text{Rec}(M(\omega)),\) the authors study also the link between the continuity of \(M\) and \(R_{M}\); then they establish properties of \(R_{M_{1}\cap M_{2}},\) \(R_{M_{1}\cup M_{2}},\) \(R_{M_{1}\times M_{2}}\) and \(R_{L\circ M}.\) Applications to marginal functions and to asymptotic directions of level sets are also given.