Tomaszewski's Problem on Randomly Signed Sums: Breaking the 3/8 Barrier.

Let v(1), v(2),...,v(n) be real numbers whose squares add up to 1. Consider the 2(n) signed sums of the form S - Sigma +/- v(i). Holzman and Kleitman (1992) proved that at least g of these sums satisfy vertical bar S vertical bar <= 1. This 3/8 bound seems to be the best their method can achieve. Using a different method, we improve the bound to 13/32, thus breaking the 3/8 barrier.