On the problem of reversibility of the entropy power inequality
  As was shown recently by the authors, the entropy power inequality can be reversed for independent summands with sufficiently concave densities, when the distributions of the summands are put in a special position. In this note it is proved that reversibility is impossible over the whole class of convex probability distributions. Related phenomena for identically distributed summands are also discussed. 
Sergey G. Bobkov and Mokshay M. Madiman
math.FA cs.IT math.IT math.PR