A short analytic proof of Fejes Tóth's theorem on sums of moments

Abstract

This article contains a simple analytic proof of the theorem of L. Fejes Tóth on sums of moments using a bare minimum of geometric arguments.Mathematics Subject Classification (1991). Primary 52C99.

A new version of l. Fejes Tóth’s moment theorem

“…which contradicts the asymptotic optimality of Ξ n for large n. Next we claim that if n is large, and Π is a DirichletVoronoi cell for Ξ n and C such that x + 6η 4 √ n B 2 ⊂ C for some x ∈ Π then (17) Π ⊂ int C is a convex polygon and diam Π 4ηn − 1 4 .…”

“…The Moment Theorem and its analogues have numerous applications in the theory of packing and covering, polytopal approximation, numerical integration, information theory, etc., (see L. Fejes Tóth [13], A. Florian [15] and P. M. Gruber [19], [20] and [21]). Knowing the profound importance, it is not surprising that numerous additional proofs are available (see G. Fejes Tóth [8], A. Florian [14] and P. M. Gruber [17]). If f is a strictly monotone increasing function of x , then G. Fejes Tóth [9] and P. M. Gruber [18] proved that the typical DirichletVoronoi cell is asymptotically a regular hexagon in any optimal conguration of at mostn points for the Moment Theorem (2).…”

An isoperimetric inequality for an integral operator on flat tori

“…Here we translate and discuss László Fejes Tóth's proof of his Moment Theorem, which was published in German [Fej72] and Russian [Fej58]. Other references for the Moment Theorem include [Fej73] where a sum of moments theorem is proved, [Gru99] where the sum of moments theorem is proved using the Moment Lemma below, [BC10] where a new version of the Moment Theorem appears with quadratic forms instead of moments, and [Gru07] which contains a very nice summary of Moment type results. See also [BPT14] for an application of the Moment Theorem to block copolymer structures.…”

“…Remark B.5. [Gru99] states the Moment Theorem for convex polygons with n = 3, 4, 5, or 6 vertices. This generalized statement follows from Proposition B.1 by introducing false vertices for n = 3, 4, or 5.…”

Hexagonal Patterns in a Simplified Model for Block Copolymers

“…Proofs in three dimensions are even rarer [29]. Also, while the hexagonal pattern appears in a wide range of situations, again there are few rigorous results [24,28,23,36].…”