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

It is shown that a recently introduced lower cone distribution function, together with the set-valued multivariate quantile, generates a Galois connection between a complete lattice of closed convex sets and the interval $[0,1]$. This generalizes the 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.