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Title: Universal optimality of the triangular lattice in the plane
Plain: In 2022 the $E8$ and Leech lattices were proved to be the best arrangements in 8 and 24 dimensions for every repulsive force from a broad class. The matching claim for the triangular lattice in two dimensions is unproved.
Precise: Prove that for every density $\rho$ the triangular lattice minimizes the energy per point $\operatorname{sum}_{y != x} f(\mid x - y \mid^{2})$ among all periodic configurations in $R^{2}$ of density $\rho$, for every completely monotone f (including $\operatorname{exp}(-a r^{2})$ and $r^{-s}$). Answer: proof or a periodic configuration beating the triangular lattice for some such f.
Settled by: A proof by an interpolation or linear-programming argument in the style of the 8D and 24D case, or an explicit counterexample.
Kind: well-posed
Answer type: proof
References: (empty)
Literature status: open
Status note: Proved for $E8$ and Leech (Cohn, Kumar, Miller, Radchenko, Viazovska, Annals 2022); in 2D known only among lattices; Petrache and Serfaty (2020) showed it would imply crystallization at next order for 2D Coulomb and Riesz gases.
Posed since: 2007