Nonregular extreme points in the set of Minkowski additive selections

Abstract:

A function $s:{\mathcal {K}^n} \to {\mathbb {R}^n}$, defined on the family ${\mathcal {K}^n}$ of all compact convex and nonempty sets in ${\mathbb {R}^n}$, is called a Minkowski additive selection, provided $s(A + B) = s(A) + s(B)$ and $s(A) \in A$, whenever $A,\;B \in {\mathcal {K}^n}$. We confirm the conjecture [6] that there exist extremal selections which are not regular ($s$ is regular if $s\left ( A \right ) \in \operatorname {ext} A,\;A \in {\mathcal {K}^n}$).