Let $V \subseteq \{0,1\}^n$, $\log|V| = k$. Consider 


$V\_r:= \bigcup\_{x \in V} V\_r(x)$, where $V\_r(x)$ is a Hamming full-ball of radius $r$ and center $x$.


What is a lower bound for the cardinality of $V\_r$? Is the cardinality of $V\_r$ the smallest when $V$ is a Hamming full-ball (as in Harper's theorem)?



Union of Hamming balls
This is Lemma 2.2 in Ahslwede-Katona
<http://www.sciencedirect.com/science/article/pii/0012365X77900176>


