## Abstract

The edge isoperimetric inequality in the discrete cube specifies, for each pair of integers m and n, the minimum size g_{n}(m) of the edge boundary of an m-element subset of (0,1)^{n}; the extremal families (up to automorphisms of the discrete cube) are initial segments of the lexicographic ordering on (0,1)^{n}. We show that for any m-element subset F ⊂ (0,1)^{n} and any integer l, if the edge boundary of F has size at most g_{n}(m)+l, then there exists an extremal family G⊂(0,1)^{n} such that |FΔG| ≤ Cl, where C is an absolute constant. This is best possible, up to the value of C. Our result can be seen as a 'stability' version of the edge isoperimetric inequality in the discrete cube, and as a discrete analogue of the seminal stability result of Fusco, Maggi and Pratelli [15] for the isoperimetric inequality in Euclidean space.

Original language | American English |
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Pages (from-to) | 1-29 |

Number of pages | 29 |

Journal | Discrete Analysis |

Volume | 9 |

Issue number | 2018 |

DOIs | |

State | Published - 2018 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2018 David Ellis, Nathan Keller and Noam Lifshitz.

## Keywords

- Discrete cube
- Discrete isoperimetric inequalities
- Stability