# On Isoperimetric Stability

@article{Lev2018OnIS, title={On Isoperimetric Stability}, author={Vsevolod F. Lev}, journal={Discrete Analysis}, year={2018} }

We show that a non-empty subset of an abelian group with a small edge boundary must be large; in particular, if $A$ and $S$ are finite, non-empty subsets of an abelian group such that $S$ is independent, and the edge boundary of $A$ with respect to $S$ does not exceed $(1-\gamma)|S||A|$ with a real $\gamma\in(0,1]$, then $|A| \ge 4^{(1-1/d)\gamma |S|}$, where $d$ is the smallest order of an element of $S$. Here the constant $4$ is best possible.
As a corollary, we derive an upper bound for the… Expand

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