This note deals with some properties of particular bounded sets w.r.t. linear continuous-time systems described by x(t) = A(o)x(t) + c(t), where c(t) is-an-element-of OMEGA subset-of R(n), OMEGA a compact set, and matrix e(tA)o) has the property of leaving a proper cone K positively invariant, that is e(tA)o K subset-of K. The considered bounded sets D(K; a, b)) are described as the intersection of shifted cones. Necessary and sufficient conditions are given. They guarantee that such sets am positively invariant w.r.t. the considered system. The trajectories starting from x(o)2 is-an-element-of R(n)D(K; a, b) (respectively x(o) is-an-element-of R(n) are studied in terms of attractivity and contractivity of the set D(K; a, b). The results are applied to the study of the constrained feedback regulator problem.