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 re\ufb01nement 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 de\ufb01ned on the subsets of a \ufb01nite set S satisfying PX\u2286S fi(X) \u2265 0 for i \u2208 {1,2}, there exists a strictly positive multiplicative set function \u00b5 over S and two subsets A,B \u2286 S such that for i \u2208 {1,2} \u00b5(A)fi(A)+\u00b5(B)fi(B)+\u00b5(A\u222aB)fi(A\u222aB)+\u00b5(A\u2229B)fi(A\u2229B)\u22650. This theorem is an analog for discrete set functions of a geometric result of Lov\u00b4asz and Simonovits.<\/p>\n","protected":false},"excerpt":{"rendered":"
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 re\ufb01nement of the 4FT is proved showing that the hypothesis of the 4FT implies a family of inequalities whose sum is the conclusion […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"MSR-TR-2003-07","msr_organization":"","msr_pages_string":"14","msr_page_range_start":"14","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"","msr_doi":"","msr_arxiv_id":"","msr_s2_paper_id":"","msr_mag_id":"","msr_pubmed_id":"","msr_other_authors":"Laszlo Lovasz, Michael 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