{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,9]],"date-time":"2026-04-09T12:38:56Z","timestamp":1775738336106,"version":"3.50.1"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,10,6]],"date-time":"2024-10-06T00:00:00Z","timestamp":1728172800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,10,6]],"date-time":"2024-10-06T00:00:00Z","timestamp":1728172800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100011603","name":"Labex B\u00e9zout","doi-asserted-by":"publisher","award":["ANR-10-LABX-58"],"award-info":[{"award-number":["ANR-10-LABX-58"]}],"id":[{"id":"10.13039\/501100011603","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2025,7]]},"DOI":"10.1007\/s00454-024-00694-2","type":"journal-article","created":{"date-parts":[[2024,10,6]],"date-time":"2024-10-06T01:01:31Z","timestamp":1728176491000},"page":"95-113","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Note on B\u00e9zout Type Inequalities for Mixed Volumes and Minkowski Sums"],"prefix":"10.1007","volume":"74","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-9865-6149","authenticated-orcid":false,"given":"Cheikh Saliou","family":"Ndiaye","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,10,6]]},"reference":[{"issue":"2","key":"694_CR1","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1023\/A:1005087216335","volume":"74","author":"S Alesker","year":"1999","unstructured":"Alesker, S., Dar, S., Milman, V.: A remarkable measure preserving diffeomorphism between two convex bodies in $$\\mathbb{R} ^n$$. Geom. Dedic. 74(2), 201\u2013212 (1999)","journal-title":"Geom. Dedic."},{"key":"694_CR2","doi-asserted-by":"publisher","first-page":"1719","DOI":"10.1007\/s00208-019-01834-3","volume":"374","author":"D Alonso-Guti\u00e9rrez","year":"2019","unstructured":"Alonso-Guti\u00e9rrez, D., Artstein-Avidan, S., Gonz\u00e1lez Merino, B., Jim\u00e9nez, C.H., Villa, R.: Rogers-Shephard and local Loomis-Whitney type inequalities. Math. Ann. 374, 1719\u20131771 (2019)","journal-title":"Math. Ann."},{"issue":"2","key":"694_CR3","doi-asserted-by":"publisher","first-page":"1350031","DOI":"10.1142\/S0219199713500314","volume":"16","author":"S Artstein-Avidan","year":"2014","unstructured":"Artstein-Avidan, S., Florentin, D., Ostrover, Y.: Remarks about mixed discriminants and volumes. Commun. Contemp. Math. 16(2), 1350031 (2014)","journal-title":"Commun. Contemp. Math."},{"issue":"3","key":"694_CR4","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1515\/advgeom-2017-0063","volume":"18","author":"S Brazitikos","year":"2018","unstructured":"Brazitikos, S., Giannopoulos, A., Liakopoulos, D.-M.: Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies. Adv. Geom. 18(3), 345\u2013354 (2018)","journal-title":"Adv. Geom."},{"issue":"2","key":"694_CR5","first-page":"183","volume":"9","author":"DN Bernstein","year":"1975","unstructured":"Bernstein, D.N.: The number of roots of a system of equations. Funct. Anal. Appl. 9(2), 183\u2013185 (1975)","journal-title":"Funct. Anal. Appl."},{"key":"694_CR6","doi-asserted-by":"publisher","first-page":"3309","DOI":"10.1016\/j.jfa.2012.01.011","volume":"262","author":"S Bobkov","year":"2012","unstructured":"Bobkov, S., Madiman, M.: Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures. J. Funct. Anal. 262, 3309\u20133339 (2012)","journal-title":"J. Funct. Anal."},{"issue":"8","key":"694_CR7","doi-asserted-by":"publisher","first-page":"2150084","DOI":"10.1142\/S021919972150084X","volume":"24","author":"KJ B\u00f6r\u00f6czky","year":"2022","unstructured":"B\u00f6r\u00f6czky, K.J., Hug, D.: Reverse Alexandrov-Fenchel inequalities for zonoids. Commun. Contemp. Math. 24(8), 2150084 (2022)","journal-title":"Commun. Contemp. Math."},{"issue":"4","key":"694_CR8","doi-asserted-by":"publisher","first-page":"375","DOI":"10.1002\/cpa.3160440402","volume":"44","author":"Y Brenier","year":"1991","unstructured":"Brenier, Y.: Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Math. 44(4), 375\u2013417 (1991)","journal-title":"Commun. Pure Appl. Math."},{"key":"694_CR9","first-page":"764","volume":"203","author":"W Fenchel","year":"1936","unstructured":"Fenchel, W.: G\u00e9n\u00e9ralisation du th\u00e9or\u00e8me de Brunn et Minkowski concernant les corps convexes. C. R. Acad. Sci. Paris. 203, 764\u2013766 (1936)","journal-title":"C. R. Acad. Sci. Paris."},{"key":"694_CR10","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1007\/BF02776055","volume":"135","author":"M Fradelizi","year":"2003","unstructured":"Fradelizi, M., Giannopoulos, A., Meyer, M.: Some inequalities about mixed volumes. Isr. J. Math. 135, 157\u2013179 (2003)","journal-title":"Isr. J. Math."},{"key":"694_CR11","first-page":"110247","volume":"286","author":"M Fradelizi","year":"2024","unstructured":"Fradelizi, M., Madiman, M., Meyer, M., Zvavitch, A.: On the volume of the Minkowski sum of zonoids. J. Math. Anal. Appl. 286, 110247 (2024)","journal-title":"J. Math. Anal. Appl."},{"key":"694_CR12","doi-asserted-by":"publisher","DOI":"10.1093\/imrn\/rnae095","author":"M Fradelizi","year":"2024","unstructured":"Fradelizi, M., Madiman, M., Zvavitch, A.: Sumsets estimates in convex geometry. Int. Math. Res. Not. (2024). https:\/\/doi.org\/10.1093\/imrn\/rnae095","journal-title":"Int. Math. Res. Not."},{"issue":"8","key":"694_CR13","doi-asserted-by":"publisher","first-page":"2403","DOI":"10.1090\/S0002-9939-02-06329-3","volume":"130","author":"A Giannopoulos","year":"2002","unstructured":"Giannopoulos, A., Hartzoulaki, M., Paouris, G.: On a local version of the Aleksandrov-Fenchel inequality about the quermasintegrals of a convex body. Proc. Am. Math. Soc. 130(8), 2403\u20132412 (2002)","journal-title":"Proc. Am. Math. Soc."},{"key":"694_CR14","doi-asserted-by":"publisher","first-page":"38","DOI":"10.1007\/BF01077562","volume":"12","author":"AG Khovanskii","year":"1978","unstructured":"Khovanskii, A.G.: Newton polyhedra and the genus of complete intersections. Funct. Anal. Appl. 12, 38\u201346 (1978)","journal-title":"Funct. Anal. Appl."},{"issue":"3","key":"694_CR15","first-page":"82","volume":"10","author":"AG Kushnirenko","year":"1976","unstructured":"Kushnirenko, A.G.: Newton polyhedra and B\u00e9zout\u2019s theorem (Russian) Funkcional. Anal. i Prilozhen 10(3), 82\u201383 (1976)","journal-title":"Anal. i Prilozhen"},{"key":"694_CR16","doi-asserted-by":"crossref","unstructured":"Lehmann, B., Xiao, J.: Correspondences between convex geometry and complex geometry. Epijournal de G\u00e9om\u00e9trie Alg\u00e9brique 15\u201317 (2017)","DOI":"10.46298\/epiga.2017.volume1.2038"},{"issue":"3","key":"694_CR17","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1023\/A:1004958110076","volume":"67","author":"IZ Ruzsa","year":"1997","unstructured":"Ruzsa, I.Z.: The Brunn-Minkowski inequality and nonconvex sets. Geom. Dedic. 67(3), 337\u2013348 (1997)","journal-title":"Geom. Dedic."},{"issue":"12","key":"694_CR18","doi-asserted-by":"publisher","first-page":"5333","DOI":"10.1090\/proc\/13149","volume":"144","author":"C Saroglou","year":"2016","unstructured":"Saroglou, C., Soprunov, I., Zvavitch, A.: Characterization of simplices via the B\u00e9zout inequality for mixed volumes. Proc. Am. Math. Soc. 144(12), 5333\u20135340 (2016)","journal-title":"Proc. Am. Math. Soc."},{"key":"694_CR19","volume-title":"Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and Its Applications","author":"R Schneider","year":"2014","unstructured":"Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 44. Cambridge University Press, Cambridge (2014)"},{"key":"694_CR20","doi-asserted-by":"crossref","unstructured":"Soprunov, I., Zvavitch, A.: B\u00e9zout inequality for mixed volumes. Int. Math. Res. Not. 7230\u20137252 (2016)","DOI":"10.1093\/imrn\/rnv390"},{"issue":"2","key":"694_CR21","doi-asserted-by":"publisher","first-page":"127461","DOI":"10.1016\/j.jmaa.2023.127461","volume":"529","author":"M Szusterman","year":"2024","unstructured":"Szusterman, M.: Extremizers in Soprunov and Zvavitch\u2019s Bezout inequalities for mixed volumes. J. Math. Anal. Appl. 529(2), 127461 (2024)","journal-title":"J. Math. Anal. Appl."},{"key":"694_CR22","volume-title":"Topics in Optimal Transportation. Graduate Studies in Mathematics","author":"C Villani","year":"2003","unstructured":"Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence (2003)"},{"key":"694_CR23","unstructured":"Xiao, J.: B\u00e9zout type inequality in convex geometry. Int. Math. Res. Not. 1\u20135 (2019)"}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-024-00694-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-024-00694-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-024-00694-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T21:14:16Z","timestamp":1750281256000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-024-00694-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,6]]},"references-count":23,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,7]]}},"alternative-id":["694"],"URL":"https:\/\/doi.org\/10.1007\/s00454-024-00694-2","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,10,6]]},"assertion":[{"value":"21 June 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 July 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 September 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 October 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}