Gauss's Law shows that the electric field everywhere inside a spherical shell of uniform charge density is $0$. Suppose we have a surface which divides space into two disjoint regions (an interior and exterior). If the electric field is $0$ everywhere interior to the surface, does it follow that the surface is a spherical shell of uniform charge density?
Edit: I am stupid. As Alfred Centauri pointed out, a zero electric field everywhere means there is no restriction on the surface. So let me impose the condition that the surface does not have charge zero.