  The main purpose of this paper is to address two open questions raised by W.
Reichel on characterizations of balls in terms of the Riesz potential and
fractional Laplacian. For a bounded $C^1$ domain $\Omega\subset \mathbb R^N$,
we consider the Riesz-potential $$u(x)=\int_{\Omega}\frac{1}{|x-y|^{N-\alpha}}
\,dy$$ for $2\leq \alpha \not =N$. We show that $u=$ constant on the boundary
of $\Omega$ if and only if $\Omega$ is a ball. In the case of $ \alpha=N$, the
similar characterization is established for the logarithmic potential. We also
prove that such a characterization holds for the logarithmic Riesz potential.
This provides a characterization for the overdetermined problem of the
fractional Laplacian. These results answer two open questions of W. Reichel to
some extent.
