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 $d>2$.

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