Local concentration inequalities and Tomaszewski's conjecture

We prove Tomaszewski’s conjecture (1986): Let f:−1,1n → ℝ be of the form f(x)= ∑i=1n ai xi. Then Pr[|f(x)| ≤ √Var[f]] ≥ 1/2. Our main novel tools are local concentration inequalities and an improved Berry-Esseen inequality for first-degree functions on the discrete cube. These tools are of independent interest, and may be useful in the study of linear threshold functions and of low degree Boolean functions.