On the complements of union of open balls of fixed radius in the Euclidean space

Abstract

Let an $R$-body be the complement of the union of open balls of radius $R$ in $\mathbb{E}^d$. The $R$-hulloid of a closed not empty set $A$, the minimal $R$-body containing $A$, is investigated; if $A$ is the set of the vertices of a simplex, the $R$-hulloid of $A$ is completely described (if $d=2$) and if $d> 2$ special examples are studied. The class of $R$-bodies is compact in the Hausdorff metric if $d =2$, but not compact if $d>2$.