{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,30]],"date-time":"2025-08-30T16:51:49Z","timestamp":1756572709582,"version":"3.41.0"},"reference-count":18,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,7,5]],"date-time":"2024-07-05T00:00:00Z","timestamp":1720137600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,7,5]],"date-time":"2024-07-05T00:00:00Z","timestamp":1720137600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2025,7]]},"abstract":"Abstract<\/jats:title>\n While studying set function properties of Lebesgue measure, F. Barthe and M. Madiman proved that Lebesgue measure is fractionally superadditive on compact sets in \n \n $$\\mathbb {R}^n$$<\/jats:tex-math>\n \n \n \n R<\/mml:mi>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. In doing this they proved a fractional generalization of the Brunn\u2013Minkowski\u2013Lyusternik (BML) inequality in dimension \n \n $$n=1$$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. In this paper we will prove the equality conditions for the fractional superadditive volume inequalites for any dimension. The non-trivial equality conditions are as follows. In the one-dimensional case we will show that for a fractional partition \n \n $$(\\mathcal {G},\\beta )$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n G<\/mml:mi>\n ,<\/mml:mo>\n \u03b2<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and nonempty sets \n \n $$A_1,\\dots ,A_m\\subseteq \\mathbb {R}$$<\/jats:tex-math>\n \n \n \n A<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u22ef<\/mml:mo>\n ,<\/mml:mo>\n \n A<\/mml:mi>\n m<\/mml:mi>\n <\/mml:msub>\n \u2286<\/mml:mo>\n R<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, equality holds iff for each \n \n $$S\\in \\mathcal {G}$$<\/jats:tex-math>\n \n \n S<\/mml:mi>\n \u2208<\/mml:mo>\n G<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, the set \n \n $$\\sum _{i\\in S}A_i$$<\/jats:tex-math>\n \n \n \n \u2211<\/mml:mo>\n \n i<\/mml:mi>\n \u2208<\/mml:mo>\n S<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n A<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is an interval. In the case of dimension \n \n $$n\\ge 2$$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n \u2265<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> we will show that equality can hold if and only if the set \n \n $$\\sum _{i=1}^{m}A_i$$<\/jats:tex-math>\n \n \n \n \u2211<\/mml:mo>\n \n i<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n m<\/mml:mi>\n <\/mml:msubsup>\n \n A<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> has measure 0.<\/jats:p>","DOI":"10.1007\/s00454-024-00672-8","type":"journal-article","created":{"date-parts":[[2024,7,5]],"date-time":"2024-07-05T17:01:41Z","timestamp":1720198901000},"page":"242-269","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Equality Conditions for the Fractional Superadditive Volume Inequalities"],"prefix":"10.1007","volume":"74","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-9044-1346","authenticated-orcid":false,"given":"Mark","family":"Meyer","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,7,5]]},"reference":[{"issue":"4","key":"672_CR1","doi-asserted-by":"publisher","first-page":"975","DOI":"10.1090\/S0894-0347-04-00459-X","volume":"17","author":"S Artstein","year":"2004","unstructured":"Artstein, S., Ball, K., Barthe, F., Naor, A.: Solution of Shannon\u2019s problem on the monotonicity of entropy. 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