A new approach to the entropy power inequality, via rearrangements

A new lower bound on the entropy of the sum of independent random vectors is demonstrated in terms of rearrangements. This lower bound is better than that given by the entropy power inequality. In fact, we use it to give a new, independent, and simple proof of the entropy power inequality in the case when the summands are identically distributed. We also give a more involved but new way to recover the full entropy power inequality, without invoking Fisher information, MMSE or any differentiation of information functionals.