A ROBUST KHINTCHINE INEQUALITY, AND ALGORITHMS FOR COMPUTING OPTIMAL CONSTANTS IN FOURIER ANALYSIS AND HIGH-DIMENSIONAL GEOMETRY

摘要

This paper makes two contributions towards determining some well-studied optimal constants in Fourier analysis of Boolean functions and high-dimensional geometry. It has been known since 1994 [C. Gotsman and N. Linial, Combinatorica, 14 (1994), pp. 35-50] that every linear threshold function (LTF) has a squared Fourier mass of at least 1/2 on its degree-0 and degree-1 coefficients. Let the minimum such Fourier mass be W-<= 1[LTF], where the minimum is taken over all n-variable LTFs and all n >= 0. Benjamini, Kalai, and Schramm [Publ. Math. Inst. Hautes Etudes Sci., 90 (1999), pp. 5-43] conjectured that the true value of W-<= 1[LTF] is 2/pi. We make progress on this conjecture by proving that W-<= 1[LTF] >= 1/2 + c for some absolute constant c > 0. The key ingredient in our proof is a "robust" version of the well-known Khintchine inequality in functional analysis, which we believe may be of independent interest. Let W-<= 1[LTFn] denote the minimum squared Fourier mass on the degree-0 and degree-1 coefficients of any n-variable LTF. We prove that for every eta > 0, there is a value K = K(eta) = poly(1/eta)such that W-<= 1[LTF] <= W-<= 1[LTFK] <= W-< 1[LTF] + eta. This easily yields an algorithm that runs in time 2(poly(1/eta)) and determines the value of W= 1[LTF] up to an additive error of +/-eta. We give an analogous structural result, and a similar 2(poly(1/eta))-time algorithm, to determine Tomaszewski's constant to within an additive error of +/-eta; this is the minimum (over all origin-centered hyperplanes H) fraction of points in {-1, 1}(n) that lie within a Euclidean distance 1 of H. Tomaszewski's constant is conjectured to be 1/2; lower bounds on it have been given by Holzman and Kleitman [Combinatorica, 12 (1992), pp. 303-316] and independently by Ben-Tal, Nemirovski, and Roos [SIAM J. Optim., 13 (2002), pp. 535-560]. Our structural results combine tools from anticoncentration of sums of independent random variables, Fourier analysis, and Hermite analysis of LTFs.