Lower cone distribution functions and set-valued quantiles form Galois connections

It is shown that the recently introduced lower cone distribution function and the associated set-valued multivariate quantile generate a Galois connection between a complete lattice of closed convex sets and the intervall [0,1]. This generalizes the (not so well-known) corresponding univariate result. It is also shown that an extension of the lower cone distribution function and the set-valued quantile characterize the capacity functional of a random set extension of the original multivariate variable along with its distribution.