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.
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