An isoperimetric inequality for the Hamming cube and some consequences
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- by Jeff Kahn and Jinyoung Park PDF
- Proc. Amer. Math. Soc. 148 (2020), 4213-4224 Request permission
Abstract:
Our basic result, an isoperimetric inequality for Hamming cube $Q_n$, can be written: \[ \int h_A^\beta d\mu \ge 2 \mu (A)(1-\mu (A)). \] Here $\mu$ is uniform measure on $V=\{0,1\}^n$ ($=V(Q_n)$); $\beta =\log _2(3/2)$; and, for $S\subseteq V$ and $x\in V$, \[ h_S(x) = \begin {cases} d_{V \setminus S}(x) &\text { if } x \in S, \\ 0 &\text { if } x \notin S \end {cases} \] (where $d_T(x)$ is the number of neighbors of $x$ in $T$).
This implies inequalities involving mixtures of edge and vertex boundaries, with related stability results, and suggests some more general possibilities. One application, a stability result for the set of edges connecting two disjoint subsets of $V$ of size roughly $|V|/2$, is a key step in showing that the number of maximal independent sets in $Q_n$ is $(1+o(1))2n\exp _2[2^{n-2}]$. This asymptotic statement, whose proof will appear separately, was the original motivation for the present work.
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Additional Information
- Jeff Kahn
- Affiliation: Department of Mathematics, Hill Center for the Mathematical Sciences, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
- MR Author ID: 96815
- Email: jkahn@math.rutgers.edu
- Jinyoung Park
- Affiliation: Department of Mathematics, Hill Center for the Mathematical Sciences, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
- MR Author ID: 1378333
- ORCID: 0000-0003-3962-5668
- Email: jp1324@math.rutgers.edu
- Received by editor(s): September 26, 2019
- Received by editor(s) in revised form: March 19, 2020
- Published electronically: July 20, 2020
- Additional Notes: The authors were supported by NSF Grant DMS1501962 and BSF Grant 2014290.
The first author was supported by a Simons Fellowship.
The second author is the corresponding author. - Communicated by: Patricia L. Hersh
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 4213-4224
- MSC (2010): Primary 05D05
- DOI: https://doi.org/10.1090/proc/15105
- MathSciNet review: 4135290