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Vertical perimeter versus horizontal perimeter (1701.00620v2)

Published 3 Jan 2017 in math.MG, cs.DS, math.CA, math.CO, and math.FA

Abstract: The discrete Heisenberg group $\mathbb{H}{\mathbb{Z}}{2k+1}$ is the group generated by $a_1,b_1,\ldots,a_k,b_k,c$, subject to the relations $[a_1,b_1]=\ldots=[a_k,b_k]=c$ and $[a_i,a_j]=[b_i,b_j]=[a_i,b_j]=[a_i,c]=[b_i,c]=1$ for every distinct $i,j\in {1,\ldots,k}$. Denote $S={a_1{\pm 1},b_1{\pm 1},\ldots,a_k{\pm 1},b_k{\pm 1}}$. The horizontal boundary of $\Omega\subset \mathbb{H}{\mathbb{Z}}{2k+1}$, denoted $\partial_{h}\Omega$, is the set of all $(x,y)\in \Omega\times (\mathbb{H}{\mathbb{Z}}{2k+1}\setminus \Omega)$ such that $x{-1}y\in S$. The horizontal perimeter of $\Omega$ is $|\partial{h}\Omega|$. For $t\in \mathbb{N}$, define $\partialt_{v} \Omega$ to be the set of all $(x,y)\in \Omega\times (\mathbb{H}{\mathsf{Z}}{2k+1}\setminus \Omega)$ such that $x{-1}y\in {ct,c{-t}}$. The vertical perimeter of $\Omega$ is defined by $|\partial{v}\Omega|= \sqrt{\sum_{t=1}\infty |\partialt_{v}\Omega|2/t2}$. It is shown here that if $k\ge 2$, then $|\partial_{v}\Omega|\lesssim \frac{1}{k} |\partial_{h}\Omega|$. The proof of this "vertical versus horizontal isoperimetric inequality" uses a new structural result that decomposes sets of finite perimeter in the Heisenberg group into pieces that admit an "intrinsic corona decomposition." This allows one to deduce an endpoint $W{1,1}\to L_2(L_1)$ boundedness of a certain singular integral operator from a corresponding lower-dimensional $W{1,2}\to L_2(L_2)$ boundedness. The above inequality has several applications, including that any embedding into $L_1$ of a ball of radius $n$ in the word metric on $\mathbb{H}_{\mathbb{Z}}{5}$ incurs bi-Lipschitz distortion that is at least a constant multiple of $\sqrt{\log n}$. It follows that the integrality gap of the Goemans--Linial semidefinite program for the Sparsest Cut Problem on inputs of size $n$ is at least a constant multiple of $\sqrt{\log n}$.

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