Given a closed set \(E\) in \(\mathbb{R}^d\) and a real number \(R>0\), the \(R\)-hulloid of \(E\) is the intersection of all closed sets, containing \(E\), complement of open balls of radius \(R\) not intersecting \(E\). An \(R\)-body, which was called \(2R\)-convex when introduced in the 50s, is a closed set which is its own \(R\)-hulloid. It has been believed that the \(R\)-hulloid of \(E\) can provide a mild regularization of a closed set and that an \(R\)-body cannot be too irregular. The authors of this paper disprove these claims and, in the process, they prove several interesting properties of \(R\)-bodies. For example, they show that the regularity of an \(R\)-body depends on dimension. If \(d=2\), a sequence of compact \(R\)-bodies converges in the Hausdorff metric to an \(R\)-body, but if \(d>2\), a sequence of compact \(R\)-bodies converges to an \((R\)-\(\epsilon\))-body, for any \(\epsilon\), \(0<\epsilon<R\), however, the limit body may not be an \(R\)-body as shown in an example. The authors also construct an example of a connected set with disconnected \(R\)-hulloid, and they study the \(R\)-hulloid of a simplex in \(\mathbb{R}^d\) obtaining a precise description on dimension \(d=2\).