The log-Brunn-Minkowski inequality

For origin-symmetric convex bodies the unit balls of finite dimensional Banach spaces) it is conjectured that there exist a family of inequalities each of which is stronger than the classical Brunn-Minkowski inequality and a family of inequalities each of which is stronger than the classical Minkowski mixed-volume inequality. It is shown that these two families of inequalities are "equivalent" in that once either of these inequalities is established, the other Must follow as a consequence. All of the conjectured inequalities are established for plane convex bodies. (C) 2012 Elsevier Inc. All rights reserved.