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

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.