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

## TL;DR

This paper demonstrates that lower cone distribution functions and set-valued multivariate quantiles form a Galois connection, generalizing univariate results and characterizing capacity functionals of random set extensions.

## Contribution

It introduces a Galois connection between convex sets and the interval [0,1] using these functions, extending univariate theory and linking to capacity functionals.

## Key findings

- Establishes a Galois connection between convex sets and [0,1].
- Generalizes univariate distribution function results.
- Characterizes capacity functionals of random set extensions.

## Abstract

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.