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

TL;DR

This paper studies the geometric properties of the complements of unions of open balls in Euclidean space, introduces the concept of R-hulloid for sets, and examines their compactness properties across different dimensions.

Contribution

It introduces the R-hulloid concept for sets in Euclidean space and characterizes its structure for simplexes, also analyzing compactness of R-bodies in various dimensions.

Findings

R-hulloid of a simplex is fully described in 2D.

Class of R-bodies is compact in 2D but not in higher dimensions.

Special examples of R-hulloid are studied for dimensions greater than 2.

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 $d>2$.