LetB1 be a ball of radiusr1 inSn (ℍn), and letB0 be a smaller ball of radiusr0 such thatB0 ⊂B1. ForSn we considerr1π. Let u be a solution of the problem- δm = 1 in Ω :=B1 /B0 vanishing on the boundary. It is shown that the associated functionalJ (Ω) is minimal if and only if the balls are concentric. It is also shown that the first Dirichlet eigenvalue of the Laplacian on Ω is maximal if and only if the balls are concentric.