Extremal Minkowski additive selections of compact convex sets

Abstract:

A function $f:{\mathcal {K}^n} \to {R^n}$, defined on the set of all compact convex sets in ${R^n}$, is a Minkowski additive selection, provided $f(K + L) = f(K) + f(L)$ and $f(K) \in K$ for all $K,L \in {\mathcal {K}^n}$. The paper deals with selections which are extremal in some sense, in particular we characterize the set of all Minkowski additive selections which have the property $f(K) \in {\text {ext}}(K)$ for all $K \in {\mathcal {K}^n}$, where ${\text {ext}}(K)$ is the set of all extreme points of $K$.