An "average distance" inequality for large subsets of the cube

In recent years “average distance” questions have been investigated by several people in graph theory and combinatorics. See, for instance, the paper of Winkler [W] as a nice introduction. In this note we present an inequality that gives good lower bounds for the average distance in Zarge subsets A of (0, 1 }“, thus making a first step toward solving an open problem stated by Ahlswede and Katona [AK, Pa