### 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 (US) |
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Pages (from-to) | 726-735 |

Number of pages | 10 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 113 |

Issue number | 4 |

DOIs | |

State | Published - May 1 2006 |

### 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

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## Cite this

Lovász, L., & Saks, M. (2006). A localization inequality for set functions.

*Journal of Combinatorial Theory. Series A*,*113*(4), 726-735. https://doi.org/10.1016/j.jcta.2005.03.011