Almost isoperimetric subsets of the discrete cube

Abstract: We show that a set $A \subset {0,1}^{n}$ with edge-boundary of size at most $|A| (\log_{2}(2^{n}/|A|) + \epsilon)$ can be made into a subcube by at most $(2 \epsilon/\log_{2}(1/\epsilon))|A|$ additions and deletions, provided $\epsilon$ is less than an absolute constant. We deduce that if $A \subset {0,1}^{n}$ has size $2^{t}$ for some $t \in \mathbb{N}$, and cannot be made into a subcube by fewer than $\delta |A|$ additions and deletions, then its edge-boundary has size at least $|A| \log_{2}(2^{n}/|A|) + |A| \delta \log_{2}(1/\delta) = 2^{t}(n-t+\delta \log_{2}(1/\delta))$, provided $\delta$ is less than an absolute constant. This is sharp whenever $\delta = 1/2^{j}$ for some $j \in {1,2,\ldots,t}$.