Proof of Tomaszewski's Conjecture on Randomly Signed Sums

Nathan Keller, Ohad Klein

We prove the following conjecture, due to Tomaszewski (1986): Let $X={\sum }_{i=1}^n{a}_i{x}_i$, where ${\sum }_i{a}_i^2=1$ and each $x_i$ is a uniformly random sign. Then $\Pr[|X|\l…