| Discrete Localization and Correlation Inequalities for Set Functions (2007) | |||||||||||||||
Abstract | |||||||||||||||
| Three theorems for set functions, closely related to the AhlswedeDaykin 4-function theorem (4FT), are proved. First, the conclusion of the 4FT is generalized to norms other than the L 1 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 f 1 , f 2 defined on the subsets of a finite set S satisfying X#S f i (X) 0 for i 2}, there exists a positive multiplicative set function over S and two subsets A, B such that for i 2} (A)f i (A)+(B)f i (B)+(A#B)f i (A#B)+ (A B)f i (A 0. This theorem is an analog for discrete set functions of a geometric result of Lovasz and Simonovits. | |||||||||||||||
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