A robust Khintchine inequality, and algorithms for computing optimal constants in Fourier analysis and high-dimensional geometry

This paper makes two contributions towards determining some well-studied optimal constants in Fourier analysis of Boolean functions and high-dimensional geometry. 1. It has been known since 1994 [GL94] that every linear threshold function has squared Fourier mass at least 1/2 on its degree-0 and degree-1 coefficients. Denote the minimum such Fourier mass by W^≤1[LTF], where the minimum is taken over all n-variable linear threshold functions and all n ≥ 0. Benjamini, Kalai and Schramm [BKS99] have conjectured that the true value of W^≤1[LTF] is 2/π. 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. 2 We give an algorithm with the following property: given any η > 0, the algorithm runs in time 2 ^poly(1/η) and determines the value of W ^≤1[LTF] up to an additive error of ±η. We give a similar 2 ^poly(1/η)-time algorithm to determine Tomaszewski's constant to within an additive error of ±η; this is the minimum (over all origin-centered hyperplanes H) fraction of points in {-1,1}ⁿ that lie within 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 [HK92] and independently by Ben-Tal, Nemirovski and Roos [BTNR02]. Our algorithms combine tools from anti-concentration of sums of independent random variables, Fourier analysis, and Hermite analysis of linear threshold functions.