Simpler proof of universal optimality for three-point lattices

Derive a simpler proof of the universal optimality of the three-point diagonal lattice in arbitrary dimensions, for example by applying Jensen's inequality instead of the linear-programming method.

Background

The paper proves that the three-point diagonal lattice is optimal for a class of convex potentials admitting an increasing affine supporting function. The proof uses the continuous LP bound and the associated Chebyshev-frequency argument.

Because the three-point case is comparatively manageable, the authors identify as unresolved whether the same optimality result can be obtained through a simpler direct argument, such as Jensen's inequality.