Characterization of balls by Riesz-potentials

For a bounded convex domain and consider the unit- density Riesz-potential . We show in this paper that u  =  const. on ∂G if and only if G is a ball. This result corresponds to a theorem of L.E. Fraenkel, where the ball is characterized by the Newtonian-potential (α = 2) of unit density being constant on ∂G. In the case α = N the kernel |x − y| ^α-N is replaced by  − log|x − y| and a similar characterization of balls is given. The proof relies on a recent variant of the moving plane method which is suitable for Green-function representations of solutions of (pseudo-)differential equations of higher-order.