On a conjecture concerning the sum of independent Rademacher random variables

It is shown that at least 50% of the probability mass of a sum of independent Rademacher random variables is within one standard deviation from its mean. This lower bound is sharp, it is much better than for instance the bound that can be obtained from application of the Chebishev inequality and the bound will have nice applications in finite sampling theory and in random walk theory. This old conjecture is of interest in itself, but has also an appealing reformulation in probability theory and in geometry.