Poppy-seed bagel theorem

In physics, the poppy-seed bagel theorem concerns interacting particles (e.g., electrons) confined to a bounded surface (or body) formula_1 when the particles repel each other pairwise with a magnitude that is proportional to the inverse distance between them raised to some positive power formula_2. In particular, this includes the Coulomb law observed in Electrostatics and Riesz potentials extensively studied in Potential theory. For formula_3 such particles, an equilibrium (stable) state, which depends on the parameter formula_2, is attained when the associated energy of the system is minimal (the so-called generalized Thomson problem). For large numbers of points, these equilibrium configurations provide a discretization of formula_1 which may or may not be nearly uniform with respect to the surface area (or volume) of formula_1. The "Poppy-seed bagel theorem" asserts that for a large class of sets formula_1, the uniformity property holds when the parameter formula_2 is larger than or equal to the dimension of the set formula_1. For example, when the points ("poppy-seeds") are confined to a torus embedded in 3-dimensions (or "surface of a bagel"), one can create a large number of points that are nearly uniformly spread on the surface by imposing a repulsion proportional to the inverse square distance between the points, or any stronger repulsion (formula_10). From a culinary perspective, to create the nearly perfect poppy-seed bagel where bites of equal size anywhere on the bagel would contain essentially the same number of poppy seeds, impose at least an inverse square distance repelling force on the seeds.
Formal definitions.
For a parameter formula_11 and an formula_3-point set formula_13, the formula_2-energy of formula_15 is defined as follows:
formula_16
For a compact set formula_1 we define its "minimal formula_3-point formula_2-energy" as 
formula_20
where the minimum is taken over all formula_3-point subsets of formula_1; i.e., formula_23. Configurations formula_15 that attain this infimum are called "formula_3-point formula_2-equilibrium configurations".
Poppy-seed bagel theorem for bodies.
We consider compact sets formula_27 with the Lebesgue measure formula_28 and formula_29. For every formula_30 fix an formula_3-point formula_2-equilibrium configuration formula_33. Set
formula_34
where formula_35 is a unit point mass at point formula_36. Under these assumptions, in the sense of weak convergence of measures,
formula_37
where formula_38 is the Lebesgue measure restricted to formula_1; i.e., formula_40.
Furthermore, it is true that
formula_41
where the constant formula_42 does not depend on the set formula_1 and, therefore,
formula_44
where formula_45 is the unit cube in formula_46.
Poppy-seed bagel theorem for manifolds.
Consider a smooth formula_47-dimensional manifold formula_1 embedded in formula_46 and denote its surface measure by formula_50. We assume formula_51. Assume formula_52
As before, for every formula_30 fix an formula_3-point formula_2-equilibrium configuration formula_33 and set
formula_57
Then, in the sense of weak convergence of measures,
formula_37
where formula_59. If formula_60 is the formula_47-dimensional Hausdorff measure, then
formula_62
where formula_63 is the volume of a d-ball.
The constant formula_42.
For formula_65, it is known that formula_66, where formula_67 is the Riemann zeta function. The following connection between the constant formula_42 and the problem of Sphere packing is known:
formula_69
where formula_70 is the volume of a p-ball and 
formula_71 where the supremum is taken over all families formula_72 of non-overlapping unit balls such that the limit
formula_73
exists.