Almost Isoperimetric Subsets of the Discrete Cube
We show that a set A ⊂ {0, 1} n with edge-boundary of size at most can be made into a subcube by at most (2ε/log 2 (1/ε))| A | additions and deletions, provided ε is less than an absolute constant. We deduce that if A ⊂ {0, 1} n has size 2 t for some t ∈ ℕ, and cannot be made into a subcube by fewer than δ| A | additions and deletions, then its edge-boundary has size at least provided δ is less than an absolute constant. This is sharp whenever δ = 1/2 j for some j ∈ {1, 2, . . ., t }.