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- W4288111165 abstract "Our basic result, an isoperimetric inequality for Hamming cube $Q_n$, can be written: [ int h_A^beta dmu 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 $Ssubseteq V$ and $xin V$, [ h_S(x) = begin{cases} d_{V setminus S}(x) &mbox{ if } x in S, 0 &mbox{ 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 $(1+o(1))2nexp_2[2^{n-2}]$. This asymptotic statement, whose proof will appear separately, was the original motivation for the present work." @default.
- W4288111165 created "2022-07-28" @default.
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- W4288111165 date "2019-09-10" @default.
- W4288111165 modified "2023-09-24" @default.
- W4288111165 title "An isoperimetric inequality for the Hamming cube and some consequences" @default.
- W4288111165 doi "https://doi.org/10.48550/arxiv.1909.04274" @default.
- W4288111165 hasPublicationYear "2019" @default.
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