Must the minimum of the gravitational potential generated by a given charge density occur at its center of mass? An idle wondering that falls out of some comments trying to clarify this question:
Suppose that you have some isolated mass distribution $\rho(\mathbf r)$, in principle with a smooth volumetric distribution, and that you're interested in the bottom of the gravity well that it generates. Is there anything that you can say, in general, about where this minimum occurs? In particular, must it coincide e.g. with the center of mass of the distribution?
 A: No, it does not. The CM is the point $\mathbf{r}_0$ that minimizes
$$\int d\mathbf{r} \, \rho(\mathbf{r}) |\mathbf{r} - \mathbf{r}_0|^2.$$
The point that minimizes the GPE instead maximizes
$$\int d\mathbf{r} \, \rho(\mathbf{r}) |\mathbf{r} - \mathbf{r}_0|^{-1}.$$
Mathematically, the CM minimizes the second moment of the distribution, while the GPE is the point that maximizes the first inverse moment. One can similarly define points that extremize any desired moment $\mu_n$. For example, 


*

*the point that minimizes $\mu_1$ is the generalization of the median to 3D space

*the point that minimizes $\mu_\infty$ is the center of the smallest sphere that contains the mass

*the point that maximizes $\mu_{-\infty}$ is the exact point where $\rho$ is largest


These are all generally different points. In general, as $n$ increases, $\mu_n$ begins to count all masses equally, while for lower $n$ the densest regions have an oversized contribution. There is a bit of a discontinuity at $\mu_0$ though, so I'm not sure how to directly compare $\mu_2$ and $\mu_{-1}$, but there's no reason for them to coincide in general. But at the least, $\mu_{-1}$ should lie "between", say, $\mu_{-2}$ and $\mu_{-\epsilon}$. 
If you want bounds on the inverse moments, or relations between them, there are presumably lots of papers on the subject, though at that point this is just a math question.
A: A quick counter-example, which came up as I was writing up the question, but which seems worth posting anyways:
Consider two uniform spherical masses $M$ and $m$, at a distance $d$ from each other. If $M$ is bigger than $m$, then the bottom of the well will occur reasonably close to the center of the bigger mass, but once you fix the masses you can, by adjusting $m$, pull the center of mass away from the interior of the big sphere.
This means that, generically, the minimum need not coincide with the center of mass.
(Though, that said, it would still be interesting to see if there are indeed things that you can say, in general, about that minimum.)
