# 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 language English (US) 726-735 10 Journal of Combinatorial Theory. Series A 113 4 https://doi.org/10.1016/j.jcta.2005.03.011 Published - May 1 2006

### Fingerprint

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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title = "A localization inequality for set functions",
abstract = "We prove the following theorem, which is an analog for discrete set functions of a geometric result of Lov{\'a}sz 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.",
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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

T1 - A localization inequality for set functions

AU - Lovász, László

AU - Saks, Michael

PY - 2006/5/1

Y1 - 2006/5/1

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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