### 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 f_{1}, f_{2} defined on the subsets of a finite set S, satisfying ∑_{X⊆S} f_{i}(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) f_{i}(A) + μ(B) f_{i}(B) + μ(A ∪ B) f_{i}(A ∪ B) + μ(A ∩ B) f_{i}(A ∩ B) ≥ 0. The Ahlswede-Daykin four function theorem can be deduced easily from this.

Original language | English |
---|---|

Pages (from-to) | 726-735 |

Number of pages | 10 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 113 |

Issue number | 4 |

DOIs | |

Publication status | Published - May 2006 |

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### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory, Series A*,

*113*(4), 726-735. https://doi.org/10.1016/j.jcta.2005.03.011

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

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series A*, vol. 113, no. 4, pp. 726-735. https://doi.org/10.1016/j.jcta.2005.03.011

}

TY - JOUR

T1 - A localization inequality for set functions

AU - Lovász, L.

AU - Saks, Michael

PY - 2006/5

Y1 - 2006/5

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

UR - http://www.scopus.com/inward/record.url?scp=33645553703&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33645553703&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2005.03.011

DO - 10.1016/j.jcta.2005.03.011

M3 - Article

AN - SCOPUS:33645553703

VL - 113

SP - 726

EP - 735

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 4

ER -