Lipschitz Continuous Selectors. Part I: Linear Selectors

We study various properties of Lipschitz continuous linear selectors on the family of all convex, nonempty and compact subsets of $\mathbb{R}^n$. In particular, it is shown that if s is such a selector then the Lipschitz constant of s can be estimated from below by the norm of $s(B^n)$, where $B^n$ is the unit ball. A notion of a parametric representation of convex bodies is introduced and illustrated with examples.