  Null quadrature domains are unbounded domains in $\R^n$ ($n \geq 2$) with
external gravitational force zero in some generalized sense. In this paper we
prove that the complement of null quadrature domain is a convex set with real
analytic boundary. We establish the quadratic growth estimate for the Schwarz
potential of a null quadrature domain which reduces our main result to Theorem
II of the paper of Caffarelli, Karp and Shahgholian (Ann. Math. 151(2000),
269-292), on the regularity of solution to the classical global free boundary
problem for Laplacian. We also show that any null quadrature domain with
non-zero upper Lebesgue density at infinity is half-space.
