A localization inequality for set functions

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 over S and two subsetsA,B ⊆ S such that for i ∈ {1, 2} (A)fi(A)+ (B)fi(B)+ (A∪B)fi(A∪ B) + (A ∩ B)fi(A ∩ B) 0. The Ahlswede–Daykin four function theorem can be deduced easily from this. © 2005 Elsevier Inc. All rights reserved.