A localization inequality for set functions

László Lovász, Michael Saks

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

We prove the following theorem, which is an analog for discrete set functions of a geometric result of Lovász and Simonovits. Given two real-valued set functions f1, f2 defined on the subsets of a finite set S, satisfying ∑X⊆S fi(X) ≥ 0 for i ∈ {1, 2}, there exists a positive multiplicative set function μ over S and two subsets A, B ⊆ S such that for i ∈ {1, 2} μ(A) fi(A) + μ(B) fi(B) + μ(A ∪ B) fi(A ∪ B) + μ(A ∩ B) fi(A ∩ B) ≥ 0. The Ahlswede-Daykin four function theorem can be deduced easily from this.

Original languageEnglish (US)
Pages (from-to)726-735
Number of pages10
JournalJournal of Combinatorial Theory. Series A
Volume113
Issue number4
DOIs
StatePublished - May 1 2006

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Set theory
Set-valued Function
Subset
Theorem
Finite Set
Multiplicative
Analogue

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Keywords

  • Discrete localization
  • Four function theorem
  • Inequalities
  • Set functions

Cite this

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A localization inequality for set functions. / Lovász, László; Saks, Michael.

In: Journal of Combinatorial Theory. Series A, Vol. 113, No. 4, 01.05.2006, p. 726-735.

Research output: Contribution to journalArticle

TY - JOUR

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AU - Lovász, László

AU - Saks, Michael

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N2 - We prove the following theorem, which is an analog for discrete set functions of a geometric result of Lovász and Simonovits. Given two real-valued set functions f1, f2 defined on the subsets of a finite set S, satisfying ∑X⊆S fi(X) ≥ 0 for i ∈ {1, 2}, there exists a positive multiplicative set function μ over S and two subsets A, B ⊆ S such that for i ∈ {1, 2} μ(A) fi(A) + μ(B) fi(B) + μ(A ∪ B) fi(A ∪ B) + μ(A ∩ B) fi(A ∩ B) ≥ 0. The Ahlswede-Daykin four function theorem can be deduced easily from this.

AB - We prove the following theorem, which is an analog for discrete set functions of a geometric result of Lovász and Simonovits. Given two real-valued set functions f1, f2 defined on the subsets of a finite set S, satisfying ∑X⊆S fi(X) ≥ 0 for i ∈ {1, 2}, there exists a positive multiplicative set function μ over S and two subsets A, B ⊆ S such that for i ∈ {1, 2} μ(A) fi(A) + μ(B) fi(B) + μ(A ∪ B) fi(A ∪ B) + μ(A ∩ B) fi(A ∩ B) ≥ 0. The Ahlswede-Daykin four function theorem can be deduced easily from this.

KW - Discrete localization

KW - Four function theorem

KW - Inequalities

KW - Set functions

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