Matches in SemOpenAlex for { <https://semopenalex.org/work/W2963766025> ?p ?o ?g. }
- W2963766025 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 $i,jin {1,ldots,k}$. Denote $S={a_1^{pm 1},b_1^{pm 1},ldots,a_k^{pm 1},b_k^{pm 1}}$. The horizontal boundary of $Omegasubset mathbb{H}_{mathbb{Z}}^{2k+1}$, denoted $partial_{h}Omega$, is the set of all $(x,y)in Omegatimes (mathbb{H}_{mathbb{Z}}^{2k+1}setminus Omega)$ such that $x^{-1}yin S$. The horizontal perimeter of $Omega$ is $|partial_{h}Omega|$. For $tin mathbb{N}$, define $partial^t_{v} Omega$ to be the set of all $(x,y)in Omegatimes (mathbb{H}_{mathsf{Z}}^{2k+1}setminus Omega)$ such that $x^{-1}yin {c^t,c^{-t}}$. The perimeter of $Omega$ is defined by $|partial_{v}Omega|= sqrt{sum_{t=1}^infty |partial^t_{v}Omega|^2/t^2}$. It is shown here that if $kge 2$, then $|partial_{v}Omega|lesssim frac{1}{k} |partial_{h}Omega|$. The proof of this vertical versus horizontal isoperimetric 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 $sqrt{log n}$." @default.
- W2963766025 created "2019-07-30" @default.
- W2963766025 creator A5052985577 @default.
- W2963766025 creator A5065907278 @default.
- W2963766025 date "2018-07-01" @default.
- W2963766025 modified "2023-10-01" @default.
- W2963766025 title "Vertical perimeter versus horizontal perimeter" @default.
- W2963766025 cites W143703132 @default.
- W2963766025 cites W1480903797 @default.
- W2963766025 cites W1530572757 @default.
- W2963766025 cites W1580461919 @default.
- W2963766025 cites W1596434689 @default.
- W2963766025 cites W1671098680 @default.
- W2963766025 cites W180116548 @default.
- W2963766025 cites W1974018476 @default.
- W2963766025 cites W1978783671 @default.
- W2963766025 cites W1980830593 @default.
- W2963766025 cites W1985582174 @default.
- W2963766025 cites W1995977886 @default.
- W2963766025 cites W1997263634 @default.
- W2963766025 cites W2004322147 @default.
- W2963766025 cites W2005090831 @default.
- W2963766025 cites W2010832227 @default.
- W2963766025 cites W2014789278 @default.
- W2963766025 cites W2016844369 @default.
- W2963766025 cites W2022596683 @default.
- W2963766025 cites W2032868343 @default.
- W2963766025 cites W2034889029 @default.
- W2963766025 cites W2035652501 @default.
- W2963766025 cites W2036322374 @default.
- W2963766025 cites W2036959865 @default.
- W2963766025 cites W2038921147 @default.
- W2963766025 cites W2040847787 @default.
- W2963766025 cites W2041493135 @default.
- W2963766025 cites W2044139543 @default.
- W2963766025 cites W2049600041 @default.
- W2963766025 cites W2054974227 @default.
- W2963766025 cites W2056878216 @default.
- W2963766025 cites W2060280149 @default.
- W2963766025 cites W2060451358 @default.
- W2963766025 cites W2060691299 @default.
- W2963766025 cites W2063491776 @default.
- W2963766025 cites W2071904195 @default.
- W2963766025 cites W2072211488 @default.
- W2963766025 cites W2073552032 @default.
- W2963766025 cites W2074411795 @default.
- W2963766025 cites W2074580548 @default.
- W2963766025 cites W2080389441 @default.
- W2963766025 cites W2085427851 @default.
- W2963766025 cites W2086924628 @default.
- W2963766025 cites W2088844265 @default.
- W2963766025 cites W2091399525 @default.
- W2963766025 cites W209159601 @default.
- W2963766025 cites W2096848739 @default.
- W2963766025 cites W2097636047 @default.
- W2963766025 cites W2103749128 @default.
- W2963766025 cites W2107881198 @default.
- W2963766025 cites W2120358419 @default.
- W2963766025 cites W2141353198 @default.
- W2963766025 cites W2147668104 @default.
- W2963766025 cites W2150148016 @default.
- W2963766025 cites W2152570014 @default.
- W2963766025 cites W2154876245 @default.
- W2963766025 cites W2158907148 @default.
- W2963766025 cites W2161719898 @default.
- W2963766025 cites W2167816765 @default.
- W2963766025 cites W2176446742 @default.
- W2963766025 cites W2314781788 @default.
- W2963766025 cites W2331602765 @default.
- W2963766025 cites W2502632701 @default.
- W2963766025 cites W2583644876 @default.
- W2963766025 cites W2602960984 @default.
- W2963766025 cites W2607237413 @default.
- W2963766025 cites W2619061098 @default.
- W2963766025 cites W2788840627 @default.
- W2963766025 cites W288635159 @default.
- W2963766025 cites W2949165676 @default.
- W2963766025 cites W2962997594 @default.
- W2963766025 cites W2963045194 @default.
- W2963766025 cites W2963279730 @default.
- W2963766025 cites W2963400539 @default.
- W2963766025 cites W2963683232 @default.
- W2963766025 cites W2963946210 @default.
- W2963766025 cites W3098011928 @default.
- W2963766025 cites W3099301101 @default.
- W2963766025 cites W3102393577 @default.
- W2963766025 cites W3103448792 @default.
- W2963766025 cites W3104961280 @default.
- W2963766025 cites W3137884736 @default.
- W2963766025 cites W4212897834 @default.
- W2963766025 cites W4213166429 @default.
- W2963766025 cites W4247212116 @default.
- W2963766025 cites W4251479607 @default.
- W2963766025 cites W4253235221 @default.
- W2963766025 cites W4254318127 @default.
- W2963766025 cites W4293860921 @default.
- W2963766025 cites W66600421 @default.
- W2963766025 doi "https://doi.org/10.4007/annals.2018.188.1.4" @default.
- W2963766025 hasPublicationYear "2018" @default.
- W2963766025 type Work @default.