Vertical perimeter versus horizontal perimeter

Assaf Naor, Robert Young

Research output: Contribution to journalArticle

Abstract

Given k ∈ N, the k'th discrete Heisenberg group, denoted H 2k+1, is the group generated by the elements a1, b1, . ., ak, bk, c, subject to the commutator relations [a1, b1] = · · · = [ak, bk] = c, while all the other pairs of elements from this generating set are required to commute, i.e., for every distinct i, j ∈ [1, . . . . k], we have [ai, aj] = [bi, bj ] = [ai, bj] = [ai, c] = [bi, c] = 1. (In particular, this implies that c is in the center of H 2k+1.) Denote b[cyrillic]k = [a1, b1, a1 -1, b1 -1, . . ., ak, bk, ak -1, bk -1]. The hori- zontal boundary of Ω ⊆ H 2k+1, denoted ∂hΩ, is the set of all those pairs (x, y) ∈ Ω ×(H 2k+1\Ω) such that x-1y ∈ b[cyrillic]k. The horizontal perimeter of Ω is the cardinality |∂hΩ| of ∂hΩ; i.e., it is the total number of edges incident to Ω in the Cayley graph induced by b[cyrillic]k. For t ∈ N, define ∂v tΩ to be the set of all those pairs (x, y) ∈ Ω × (H 2k+1\Ω) such that x-1y ∈ [ct, c-t]. Thus, |∂v tΩ| is the total number of edges incident to Ω in the (disconnected) Cayley graph induced by [ct, c-t] ⊆ H 2k+1. The vertical perimeter of Ω is defined by |∂vjΩ|. It is shown here that if k ≥ 2, then |∂vΩ|≲ |∂vΩ| The proof of this \vertical versus horizontal isoperi- metric inequality" uses a new structural result that decomposes sets of finite perimeter in the Heisenberg group into pieces that admit an \in- trinsic corona decomposition. "This allows one to deduce an endpoint W1,2 → L2(L1) boundedness of a certain singular integral operator from a corresponding lower-dimensional W1,2 → L2(L2) boundedness. Apart from its intrinsic geometric interest, the above (sharp) isoperimetric-type inequality has several (sharp) applications, including that for every n ∈ N, any embedding into an L1(μ) space of a ball of radius n in the word metric on H 5 that is induced by the generating set b[cyrillic]2 incurs bi-Lipschitz distortion that is at least a universal constant multiple of √ log n. As an application to approximation algorithms, it follows that for every n ∈ ℕ, the integral- ity gap of the Goemans-Linial semidefinite program for the Sparsest Cut Problem on inputs of size n is at least a universal constant multiple of √ log n.

Original languageEnglish (US)
Pages (from-to)171-279
Number of pages109
JournalAnnals of Mathematics
Volume188
Issue number1
DOIs
StatePublished - Jul 1 2018

Fingerprint

Perimeter
Isoperimetric
Horizontal
Generating Set
Vertical
Cayley Graph
Heisenberg Group
Boundedness
Decompose
Semidefinite Program
Corona
Singular Integral Operator
Discrete Group
Commute
Commutator
Lipschitz
Approximation Algorithms
Deduce
Cardinality
Ball

Keywords

  • Approximation algorithms
  • Heisenberg group
  • Isoperimetric inequalities
  • Metric embeddings
  • Metrics of negative type
  • Semidefinite programming
  • Sparsest Cut Problem

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

Vertical perimeter versus horizontal perimeter. / Naor, Assaf; Young, Robert.

In: Annals of Mathematics, Vol. 188, No. 1, 01.07.2018, p. 171-279.

Research output: Contribution to journalArticle

Naor, Assaf ; Young, Robert. / Vertical perimeter versus horizontal perimeter. In: Annals of Mathematics. 2018 ; Vol. 188, No. 1. pp. 171-279.
@article{c0890faa88c04714bbf436dd14087b99,
title = "Vertical perimeter versus horizontal perimeter",
abstract = "Given k ∈ N, the k'th discrete Heisenberg group, denoted Hℤ 2k+1, is the group generated by the elements a1, b1, . ., ak, bk, c, subject to the commutator relations [a1, b1] = · · · = [ak, bk] = c, while all the other pairs of elements from this generating set are required to commute, i.e., for every distinct i, j ∈ [1, . . . . k], we have [ai, aj] = [bi, bj ] = [ai, bj] = [ai, c] = [bi, c] = 1. (In particular, this implies that c is in the center of Hℤ 2k+1.) Denote b[cyrillic]k = [a1, b1, a1 -1, b1 -1, . . ., ak, bk, ak -1, bk -1]. The hori- zontal boundary of Ω ⊆ Hℤ 2k+1, denoted ∂hΩ, is the set of all those pairs (x, y) ∈ Ω ×(Hℤ 2k+1\Ω) such that x-1y ∈ b[cyrillic]k. The horizontal perimeter of Ω is the cardinality |∂hΩ| of ∂hΩ; i.e., it is the total number of edges incident to Ω in the Cayley graph induced by b[cyrillic]k. For t ∈ N, define ∂v tΩ to be the set of all those pairs (x, y) ∈ Ω × (Hℤ 2k+1\Ω) such that x-1y ∈ [ct, c-t]. Thus, |∂v tΩ| is the total number of edges incident to Ω in the (disconnected) Cayley graph induced by [ct, c-t] ⊆ Hℤ 2k+1. The vertical perimeter of Ω is defined by |∂vjΩ|. It is shown here that if k ≥ 2, then |∂vΩ|≲ |∂vΩ| The proof of this \vertical versus horizontal isoperi- metric inequality{"} uses a new structural result that decomposes sets of finite perimeter in the Heisenberg group into pieces that admit an \in- trinsic corona decomposition. {"}This allows one to deduce an endpoint W1,2 → L2(L1) boundedness of a certain singular integral operator from a corresponding lower-dimensional W1,2 → L2(L2) boundedness. Apart from its intrinsic geometric interest, the above (sharp) isoperimetric-type inequality has several (sharp) applications, including that for every n ∈ N, any embedding into an L1(μ) space of a ball of radius n in the word metric on Hℤ 5 that is induced by the generating set b[cyrillic]2 incurs bi-Lipschitz distortion that is at least a universal constant multiple of √ log n. As an application to approximation algorithms, it follows that for every n ∈ ℕ, the integral- ity gap of the Goemans-Linial semidefinite program for the Sparsest Cut Problem on inputs of size n is at least a universal constant multiple of √ log n.",
keywords = "Approximation algorithms, Heisenberg group, Isoperimetric inequalities, Metric embeddings, Metrics of negative type, Semidefinite programming, Sparsest Cut Problem",
author = "Assaf Naor and Robert Young",
year = "2018",
month = "7",
day = "1",
doi = "10.4007/annals.2018.188.1.4",
language = "English (US)",
volume = "188",
pages = "171--279",
journal = "Annals of Mathematics",
issn = "0003-486X",
publisher = "Princeton University Press",
number = "1",

}

TY - JOUR

T1 - Vertical perimeter versus horizontal perimeter

AU - Naor, Assaf

AU - Young, Robert

PY - 2018/7/1

Y1 - 2018/7/1

N2 - Given k ∈ N, the k'th discrete Heisenberg group, denoted Hℤ 2k+1, is the group generated by the elements a1, b1, . ., ak, bk, c, subject to the commutator relations [a1, b1] = · · · = [ak, bk] = c, while all the other pairs of elements from this generating set are required to commute, i.e., for every distinct i, j ∈ [1, . . . . k], we have [ai, aj] = [bi, bj ] = [ai, bj] = [ai, c] = [bi, c] = 1. (In particular, this implies that c is in the center of Hℤ 2k+1.) Denote b[cyrillic]k = [a1, b1, a1 -1, b1 -1, . . ., ak, bk, ak -1, bk -1]. The hori- zontal boundary of Ω ⊆ Hℤ 2k+1, denoted ∂hΩ, is the set of all those pairs (x, y) ∈ Ω ×(Hℤ 2k+1\Ω) such that x-1y ∈ b[cyrillic]k. The horizontal perimeter of Ω is the cardinality |∂hΩ| of ∂hΩ; i.e., it is the total number of edges incident to Ω in the Cayley graph induced by b[cyrillic]k. For t ∈ N, define ∂v tΩ to be the set of all those pairs (x, y) ∈ Ω × (Hℤ 2k+1\Ω) such that x-1y ∈ [ct, c-t]. Thus, |∂v tΩ| is the total number of edges incident to Ω in the (disconnected) Cayley graph induced by [ct, c-t] ⊆ Hℤ 2k+1. The vertical perimeter of Ω is defined by |∂vjΩ|. It is shown here that if k ≥ 2, then |∂vΩ|≲ |∂vΩ| The proof of this \vertical versus horizontal isoperi- metric inequality" uses a new structural result that decomposes sets of finite perimeter in the Heisenberg group into pieces that admit an \in- trinsic corona decomposition. "This allows one to deduce an endpoint W1,2 → L2(L1) boundedness of a certain singular integral operator from a corresponding lower-dimensional W1,2 → L2(L2) boundedness. Apart from its intrinsic geometric interest, the above (sharp) isoperimetric-type inequality has several (sharp) applications, including that for every n ∈ N, any embedding into an L1(μ) space of a ball of radius n in the word metric on Hℤ 5 that is induced by the generating set b[cyrillic]2 incurs bi-Lipschitz distortion that is at least a universal constant multiple of √ log n. As an application to approximation algorithms, it follows that for every n ∈ ℕ, the integral- ity gap of the Goemans-Linial semidefinite program for the Sparsest Cut Problem on inputs of size n is at least a universal constant multiple of √ log n.

AB - Given k ∈ N, the k'th discrete Heisenberg group, denoted Hℤ 2k+1, is the group generated by the elements a1, b1, . ., ak, bk, c, subject to the commutator relations [a1, b1] = · · · = [ak, bk] = c, while all the other pairs of elements from this generating set are required to commute, i.e., for every distinct i, j ∈ [1, . . . . k], we have [ai, aj] = [bi, bj ] = [ai, bj] = [ai, c] = [bi, c] = 1. (In particular, this implies that c is in the center of Hℤ 2k+1.) Denote b[cyrillic]k = [a1, b1, a1 -1, b1 -1, . . ., ak, bk, ak -1, bk -1]. The hori- zontal boundary of Ω ⊆ Hℤ 2k+1, denoted ∂hΩ, is the set of all those pairs (x, y) ∈ Ω ×(Hℤ 2k+1\Ω) such that x-1y ∈ b[cyrillic]k. The horizontal perimeter of Ω is the cardinality |∂hΩ| of ∂hΩ; i.e., it is the total number of edges incident to Ω in the Cayley graph induced by b[cyrillic]k. For t ∈ N, define ∂v tΩ to be the set of all those pairs (x, y) ∈ Ω × (Hℤ 2k+1\Ω) such that x-1y ∈ [ct, c-t]. Thus, |∂v tΩ| is the total number of edges incident to Ω in the (disconnected) Cayley graph induced by [ct, c-t] ⊆ Hℤ 2k+1. The vertical perimeter of Ω is defined by |∂vjΩ|. It is shown here that if k ≥ 2, then |∂vΩ|≲ |∂vΩ| The proof of this \vertical versus horizontal isoperi- metric inequality" uses a new structural result that decomposes sets of finite perimeter in the Heisenberg group into pieces that admit an \in- trinsic corona decomposition. "This allows one to deduce an endpoint W1,2 → L2(L1) boundedness of a certain singular integral operator from a corresponding lower-dimensional W1,2 → L2(L2) boundedness. Apart from its intrinsic geometric interest, the above (sharp) isoperimetric-type inequality has several (sharp) applications, including that for every n ∈ N, any embedding into an L1(μ) space of a ball of radius n in the word metric on Hℤ 5 that is induced by the generating set b[cyrillic]2 incurs bi-Lipschitz distortion that is at least a universal constant multiple of √ log n. As an application to approximation algorithms, it follows that for every n ∈ ℕ, the integral- ity gap of the Goemans-Linial semidefinite program for the Sparsest Cut Problem on inputs of size n is at least a universal constant multiple of √ log n.

KW - Approximation algorithms

KW - Heisenberg group

KW - Isoperimetric inequalities

KW - Metric embeddings

KW - Metrics of negative type

KW - Semidefinite programming

KW - Sparsest Cut Problem

UR - http://www.scopus.com/inward/record.url?scp=85048631868&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85048631868&partnerID=8YFLogxK

U2 - 10.4007/annals.2018.188.1.4

DO - 10.4007/annals.2018.188.1.4

M3 - Article

VL - 188

SP - 171

EP - 279

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 1

ER -