Discrete Localization and Correlation Inequalities for Set Functions

Three theorems for set functions, closely related to the Ahlswede-Daykin 4-function theorem (4FT), are proved. First, the conclusion of the 4FT is generalized to norms other than the L1 norm. Secondly, a refinement of the 4FT is proved showing that the hypothesis of the 4FT implies a family of inequalities whose sum is the conclusion of the 4FT. Finally, it is also shown that the hypothesis of the four function theorem is preserved under a form of convolution. All of these theorems are deduced from another theorem proven here: 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 strictly 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. This theorem is an analog for discrete set functions of a geometric result of Lovász and Simonovits.