On the complements of union of open balls and related set classes

TL;DR

This paper investigates the properties of $R$-bodies, which are complements of unions of open balls of radius $R$, and explores their relationships with other geometric set classes like $R$-reach and $R$-rolling sets, extending convex set concepts.

Contribution

It introduces new properties of $R$-bodies and related sets, generalizing convex set properties using $R$-cones and comparing different geometric families.

Findings

Properties of $R$-bodies similar to convex sets are established.

Relationships between $R$-bodies, $R$-reach, and $R$-rolling sets are analyzed.

New properties using $R$-cones are proved for these families.

Abstract

Let a $R$-body be a closed set, complement of union of open balls of radius $R$ in the Euclidean space. Properties generalizing similar ones for convex sets are proved for the family of $R$-bodies; properties for the family of sets supported by spheres of radius $R$ (extension of the supporting hyperplane to convex bodies) are investigated. Comparison of that family with the sets of reach $R$ and with the $R$-rolling sets are studied. New properties for the previous families are proved, by using the $R$-cones, generalization of the convex cones.