Quantiles and Convexity

A real random variable admits median(s) and quantiles. These values minimize convex functions on the reals $\mathbb{R}$. This is known. We discuss these results and their relationship with some notions about functions of bounded variation developed by J. J. Moreau in his mathematical work in view of the mechanical phenomena which are shocks and friction, especially filled-in graphs and Stieltjes measures of products of BV functions.