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I'm a physicist interested in the conformal bootstrap, one version of which was recently shown to have many similarities to the problem of sphere packing. Sphere packing in $\mathbf{R}^d$ has been solved in $d=8$ and $24$, and recently those solutions were shown to be universally optimal among point configurations, i.e., "they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians)."

I'm trying to appreciate the difference between sphere packing and universal optimality-- to this end, does anyone know of a simple example for which it is clear that the densest sphere packing is not universally optimal?

My initial intuition was that sphere packings would be necessarily universally optimal since you could imagine a spherical equipotential surface surrounding each particle, and that minimizing the energy of the configuration would amount to finding the optimal packing of these equipotential surfaces. Evidently this is false.

The paper with the aforementioned proof mentions that it was found in 3 dimensions that the conjectured optimal lattice solutions for potential functions of the form $r \mapsto \mathrm{e}^{- \pi r^2}$ are not optimal when nonlattice configurations are considered, but I am hoping for a more obvious illustration.

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In three dimensions you don’t need to go beyond lattices to see the failure of universal optimality. When the potential function is sufficiently steep (e.g., a narrow Gaussian), the face-centered cubic lattice is optimal, but for wide Gaussians the body-centered cubic beats it. You can see this using Poisson summation (the face-centered and body-centered lattices are duals, so if one wins for narrow Gaussians its dual must win for wide Gaussians) or just by direct calculation.

Your intuition seems reasonable for very steep potential functions, where the energy is dominated by nearby particles, but when longer-range interactions need to be taken into account there’s no reason dense sphere packing should be optimal.

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    $\begingroup$ Also 5 points on a 2-sphere, right? Possibly a better example because the optimal packing is much easier to prove then in 3-space. (Also it should generalize to higher dimensions.) $\endgroup$ – Noam D. Elkies May 19 at 20:55

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