Almost Isoperimetric Subsets of the Discrete Cube
David C. P. Ellis · Combinatorics Probability Computing · 2011
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/log2(1/ε))|A| additions and deletions, provided ε is less than an absolute constant. We deduce that if A ⊂ {0, 1}n has size 2t 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 = 1/2j for some j ∈ {1, 2, . . ., t}.