{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,30]],"date-time":"2026-05-30T20:00:40Z","timestamp":1780171240310,"version":"3.54.0"},"reference-count":52,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T00:00:00Z","timestamp":1751587200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T00:00:00Z","timestamp":1751587200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2026,6]]},"DOI":"10.1007\/s00454-025-00740-7","type":"journal-article","created":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T17:43:22Z","timestamp":1751651002000},"page":"1358-1377","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Orlicz Gaussian Minkowski Problem for General Measures"],"prefix":"10.1007","volume":"75","author":[{"given":"Hailin","family":"Jin","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-1881-4064","authenticated-orcid":false,"given":"Dandan","family":"Lai","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Suwei","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,7,4]]},"reference":[{"key":"740_CR1","first-page":"27","volume":"3","author":"AD Aleksandrov","year":"1938","unstructured":"Aleksandrov, A.D.: On the theory of mixed volumes. III. Extension of two theorems of minkowski on convex polyhedra to arbitrary convex bodies. Mat. Sb. (NS) 3, 27\u201346 (1938)","journal-title":"Mat. Sb. (NS)"},{"key":"740_CR2","first-page":"167","volume":"6","author":"AD Aleksandrov","year":"1939","unstructured":"Aleksandrov, A.D.: On the surface area measure of convex bodies. Mat. Sbornik N. S. 6, 167\u2013174 (1939)","journal-title":"Mat. Sbornik N. S."},{"key":"740_CR3","first-page":"195","volume":"36","author":"AD Aleksandrov","year":"1942","unstructured":"Aleksandrov, A.D.: Smoothness of convex surface of bounded Gaussian curvature. C. R. (Dokl) Acad. Sci 36, 195\u2013199 (1942)","journal-title":"C. R. (Dokl) Acad. Sci"},{"key":"740_CR4","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1007\/BF02573986","volume":"4","author":"K Ball","year":"1993","unstructured":"Ball, K.: The reverse isoperimetric problem for Gaussian measure. Discrete Comput. Geom. 4, 411\u2013420 (1993)","journal-title":"Discrete Comput. Geom."},{"key":"740_CR5","doi-asserted-by":"publisher","first-page":"831","DOI":"10.1090\/S0894-0347-2012-00741-3","volume":"26","author":"KJ B\u00f6r\u00f6czky","year":"2013","unstructured":"B\u00f6r\u00f6czky, K.J., Lutwak, E., Yang, D., Zhang, G.: The logarithmic Minkowski Problem. J. Am. Math. Soc. 26, 831\u2013852 (2013)","journal-title":"J. Am. Math. Soc."},{"key":"740_CR6","doi-asserted-by":"publisher","first-page":"1406","DOI":"10.1002\/cpa.21898","volume":"73","author":"KJ B\u00f6r\u00f6czky","year":"2020","unstructured":"B\u00f6r\u00f6czky, K.J., Lutwak, E., Yang, D., Zhang, G., Zhao, Y.: The Guass image problem. Comm. Pure Appl. Math. 73, 1406\u20131452 (2020)","journal-title":"Comm. Pure Appl. Math."},{"key":"740_CR7","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1016\/j.aim.2004.11.007","volume":"201","author":"W Chen","year":"2006","unstructured":"Chen, W.: $$L_{p}$$ Minkowski problem with not necessarily positive data. Adv. Math. 201, 77\u201389 (2006)","journal-title":"Adv. Math."},{"key":"740_CR8","doi-asserted-by":"publisher","first-page":"58","DOI":"10.1016\/j.jde.2017.06.007","volume":"263","author":"S Chen","year":"2017","unstructured":"Chen, S., Li, Q., Zhu, G.: On the $$L_p$$ Monge-Ampere equation. J. Differ. Equ. 263, 58\u201381 (2017)","journal-title":"J. Differ. Equ."},{"key":"740_CR9","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2023.109351","volume":"432","author":"S Chen","year":"2023","unstructured":"Chen, S., Hu, S., Liu, W., Zhao, Y.: On the planar Gaussian\u2013Minkowski problem. Adv. Math. 432, 109351 (2023)","journal-title":"Adv. Math."},{"key":"740_CR10","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1016\/j.jmaa.2024.128656","volume":"541","author":"B Chen","year":"2025","unstructured":"Chen, B., Shi, W., Wang, W.: An inverse Gauss curvature flow to the $$L_p$$-Gauss Minkowski problem. J. Math. Anal. Appl. 541, 19 (2025)","journal-title":"J. Math. Anal. Appl."},{"key":"740_CR11","doi-asserted-by":"publisher","first-page":"281","DOI":"10.7146\/math.scand.a-12035","volume":"53","author":"A Ehrhard","year":"1983","unstructured":"Ehrhard, A.: Sym\u00e9trisation dans l\u2019espace de Gauss. Math. Scand. 53, 281\u2013301 (1983)","journal-title":"Math. Scand."},{"key":"740_CR12","first-page":"149","volume":"22","author":"A Ehrhard","year":"1986","unstructured":"Ehrhard, A.: \u00c9lements extr\u00e9maux pours les in\u00e9galit\u00e9s de Brunn-Minkowski gaussienes. Ann. Inst. H. Poincare Probab. Statist. 22, 149\u2013168 (1986)","journal-title":"Ann. Inst. H. Poincare Probab. Statist."},{"key":"740_CR13","first-page":"1","volume":"16","author":"W Fenchel","year":"1938","unstructured":"Fenchel, W., Jessen, B.: Mengenfunktionen und konvexe K\u00f6rper, Danske Vid. Selskab. Mat.-fys. Medd. 16, 1\u201331 (1938)","journal-title":"Selskab. Mat.-fys. Medd."},{"key":"740_CR14","doi-asserted-by":"publisher","first-page":"350","DOI":"10.1016\/j.jde.2023.03.026","volume":"363","author":"Y Feng","year":"2023","unstructured":"Feng, Y., Hu, S., Xu, L.: On the $$L_p$$ Gaussian Minkowski problem. J. Differ. Equ. 363, 350\u2013390 (2023)","journal-title":"J. Differ. Equ."},{"key":"740_CR15","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1007\/s12220-022-01139-z","volume":"33","author":"Y Feng","year":"2023","unstructured":"Feng, Y., Liu, W., Xu, L.: Existence of non-symmetric solutions to the Gaussian Minkowski problem. J. Geom. Anal. 33, 89 (2023)","journal-title":"J. Geom. Anal."},{"key":"740_CR16","doi-asserted-by":"publisher","first-page":"17","DOI":"10.7146\/math.scand.a-10510","volume":"10","author":"WJ Firey","year":"1962","unstructured":"Firey, W.J.: p-means of convex bodies. Math. Scand. 10, 17\u201324 (1962)","journal-title":"Math. Scand."},{"key":"740_CR17","doi-asserted-by":"publisher","first-page":"427","DOI":"10.4310\/jdg\/1406033976","volume":"97","author":"RJ Gardner","year":"2014","unstructured":"Gardner, R.J., Hug, D., Weil, W.: The Orlicz-Brunn-Minkowski theory: a general framework, additions, and inequalities. J. Differ. Geom. 97, 427\u2013476 (2014)","journal-title":"J. Differ. Geom."},{"key":"740_CR18","doi-asserted-by":"publisher","first-page":"810","DOI":"10.1016\/j.jmaa.2015.05.016","volume":"430","author":"RJ Gardner","year":"2015","unstructured":"Gardner, R.J., Hug, D., Weil, W., Ye, D.: The dual Orlicz-Brunn-Minkowski theory. J. Math. Anal. Appl. 430, 810\u2013829 (2015)","journal-title":"J. Math. Anal. Appl."},{"key":"740_CR19","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1007\/s00526-018-1449-0","volume":"58","author":"RJ Gardner","year":"2019","unstructured":"Gardner, R.J., Hug, D., Weil, W., Xing, S., Ye, D.: General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski problem I. Calc. Var. Partial. Differ. Equ. 58, 12\u201335 (2019)","journal-title":"Calc. Var. Partial. Differ. Equ."},{"key":"740_CR20","doi-asserted-by":"publisher","first-page":"2485","DOI":"10.1016\/j.aim.2010.02.006","volume":"224","author":"C Haberl","year":"2010","unstructured":"Haberl, C., Lutwak, E., Yang, D., Zhang, G.: The even Orlicz Minkowski problem. Adv. Math. 224, 2485\u20132510 (2010)","journal-title":"Adv. Math."},{"key":"740_CR21","unstructured":"Hu, J.: The generalized Gaussian log-Minkowski problem, Preprint at arXiv:2401.08427"},{"key":"740_CR22","doi-asserted-by":"publisher","first-page":"282","DOI":"10.1007\/s00454-012-9434-4","volume":"48","author":"Q Huang","year":"2012","unstructured":"Huang, Q., He, B.: On the Orlicz problem for polytopes. Discrete Comput. Geom. 48, 282\u2013297 (2012)","journal-title":"Discrete Comput. Geom."},{"key":"740_CR23","doi-asserted-by":"publisher","first-page":"930","DOI":"10.3934\/dcdss.2023118","volume":"17","author":"Y Huang","year":"2024","unstructured":"Huang, Y., Qin, L.: The Gaussian chord Minkowski problem. Discrete Conti. Dyn. Syst. Ser. S 17, 930\u2013944 (2024)","journal-title":"Discrete Conti. Dyn. Syst. Ser. S"},{"key":"740_CR24","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1007\/s11511-016-0140-6","volume":"216","author":"Y Huang","year":"2016","unstructured":"Huang, Y., Lutwak, E., Yang, D., Zhang, G.: Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems. Acta Math. 216, 325\u2013388 (2016)","journal-title":"Acta Math."},{"key":"740_CR25","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2021.107769","volume":"385","author":"Y Huang","year":"2021","unstructured":"Huang, Y., Xi, D., Zhao, Y.: The Minkowski problem in Gaussian probability space. Adv. Math. 385, 107769 (2021)","journal-title":"Adv. Math."},{"key":"740_CR26","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1007\/s00454-004-1149-8","volume":"33","author":"D Hug","year":"2005","unstructured":"Hug, D., Lutwak, E., Yang, D., Zhang, G.: On the $$L_{p}$$ Minkowski problem for polytopes. Discrete Comput. Geom. 33, 699\u2013715 (2005)","journal-title":"Discrete Comput. Geom."},{"key":"740_CR27","doi-asserted-by":"publisher","first-page":"262","DOI":"10.1016\/j.aim.2019.01.004","volume":"344","author":"H Jian","year":"2019","unstructured":"Jian, H., Lu, J.: Existence of solutions to the Orlicz-Minkowski problem. Adv. Math. 344, 262\u2013288 (2019)","journal-title":"Adv. Math."},{"key":"740_CR28","first-page":"8447","volume":"376","author":"L Kryvonos","year":"2023","unstructured":"Kryvonos, L., Langharst, D.: Weighted Minkowski\u2019s existence theorem and projection bodies. Trans. Am. Math. Soc. 376, 8447\u20138493 (2023)","journal-title":"Trans. Am. Math. Soc."},{"key":"740_CR29","doi-asserted-by":"crossref","unstructured":"Langharst, D., Liu, J., Tang, S.: The weighted $$L_p$$ Minkowski problem, Preprint at arXiv:2407.20064, (2024)","DOI":"10.2139\/ssrn.5025236"},{"key":"740_CR30","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1007\/s10711-013-9829-2","volume":"168","author":"A Li","year":"2014","unstructured":"Li, A.: The generalization of Minkowski problems for polytopes. Geom. Dedicata. 168, 245\u2013264 (2014)","journal-title":"Geom. Dedicata."},{"key":"740_CR31","doi-asserted-by":"publisher","DOI":"10.1016\/j.aam.2025.102839","volume":"165","author":"S Li","year":"2025","unstructured":"Li, S., Chen, Q., Jin, H.: The Orlicz chord Minkowski problem for general measures. Adv. Appl. Math. 165, 102839 (2025)","journal-title":"Adv. Appl. Math."},{"key":"740_CR32","unstructured":"Liu, W.: On the planar $$L_p$$-Gaussian-Minkowski problem for $$0\\le p<1$$, Preprint at arXiv:2405.13725"},{"key":"740_CR33","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1007\/s00526-021-02141-z","volume":"61","author":"J Liu","year":"2022","unstructured":"Liu, J.: The $$L_{p}$$-Gaussian Minkowski problem. Calc. Var. Partial. Differ. Equ. 61, 28 (2022)","journal-title":"Calc. Var. Partial. Differ. Equ."},{"key":"740_CR34","doi-asserted-by":"publisher","first-page":"302","DOI":"10.1007\/s12220-024-01748-w","volume":"34","author":"J Liu","year":"2024","unstructured":"Liu, J., Tang, S.: The generalized Gaussian Minkowski problem. J. Geom. Anal. 34, 302 (2024)","journal-title":"J. Geom. Anal."},{"key":"740_CR35","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2019.106803","volume":"356","author":"GV Livshyts","year":"2019","unstructured":"Livshyts, G.V.: An extension of Minkowski\u2019s theorem and its applications to questions about projections for measures. Adv. Math. 356, 106803 (2019)","journal-title":"Adv. Math."},{"key":"740_CR36","doi-asserted-by":"publisher","first-page":"131","DOI":"10.4310\/jdg\/1214454097","volume":"38","author":"E Lutwak","year":"1993","unstructured":"Lutwak, E.: The Brunn-Minkowski-Firey theory I: mixed volumes and the Minkowski problem. J. Differ. Geom. 38, 131\u2013150 (1993)","journal-title":"J. Differ. Geom."},{"key":"740_CR37","doi-asserted-by":"publisher","first-page":"244","DOI":"10.1006\/aima.1996.0022","volume":"118","author":"E Lutwak","year":"1996","unstructured":"Lutwak, E.: The Brunn-Minkowski-Firey theory II: affine and geominimal surface areas. Adv. Math. 118, 244\u2013294 (1996)","journal-title":"Adv. Math."},{"key":"740_CR38","first-page":"4359","volume":"11","author":"E Lutwak","year":"2004","unstructured":"Lutwak, E., Yang, D., Zhang, G.: On the $$L_{p}$$-Minkowski problem. Trans. Amer. Math. Soc. 11, 4359\u20134370 (2004)","journal-title":"Trans. Amer. Math. Soc."},{"key":"740_CR39","doi-asserted-by":"publisher","first-page":"220","DOI":"10.1016\/j.aim.2009.08.002","volume":"223","author":"E Lutwak","year":"2010","unstructured":"Lutwak, E., Yang, D., Zhang, G.: Orlicz projection bodies. Adv. Math. 223, 220\u2013242 (2010)","journal-title":"Adv. Math."},{"key":"740_CR40","doi-asserted-by":"publisher","first-page":"365","DOI":"10.4310\/jdg\/1274707317","volume":"84","author":"E Lutwak","year":"2010","unstructured":"Lutwak, E., Yang, D., Zhang, G.: Orlicz centroid bodies. J. Differential Geom. 84, 365\u2013387 (2010)","journal-title":"J. Differential Geom."},{"key":"740_CR41","first-page":"198","volume":"84","author":"H Minkowski","year":"1897","unstructured":"Minkowski, H.: Allgemeine Lehrs\u00e4tze \u00fcber die konvexen Polyeder. Nachr. Ges. Wiss. G\u00f6ttingen 84, 198\u2013219 (1897)","journal-title":"Nachr. Ges. Wiss. G\u00f6ttingen"},{"key":"740_CR42","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1007\/978-3-540-36428-3_15","volume":"1807","author":"F Nazarov","year":"2003","unstructured":"Nazarov, F.: On the maximal perimeter of a convex set in $$\\mathbb{R} ^{n}$$ with respect to a Gaussian measure, geometric aspects of functional analysis. Lecture Notes Math. 1807, 169\u2013187 (2003)","journal-title":"Lecture Notes Math."},{"key":"740_CR43","volume-title":"Convex bodies: the Brunn-Minkowski theory","author":"R Schneider","year":"2014","unstructured":"Schneider, R.: Convex bodies: the Brunn-Minkowski theory, 2nd edn. Cambridge University Press, Cambridge (2014)","edition":"2"},{"key":"740_CR44","unstructured":"Sheng, W., Xue, K.: Flow by Gauss curvature to the $$L_p$$-Gaussian Minkowski problem, Preprint at arXiv:2212.01822"},{"key":"740_CR45","doi-asserted-by":"publisher","first-page":"5381","DOI":"10.1090\/proc\/16989","volume":"152","author":"S Tang","year":"2024","unstructured":"Tang, S.: Existence of Solutions to $$L_p$$-Gaussian Minkowski problem. Proc. Amer. Math. Soc. 152, 5381\u20135394 (2024)","journal-title":"Proc. Amer. Math. Soc."},{"key":"740_CR46","doi-asserted-by":"publisher","first-page":"2253","DOI":"10.1007\/s10114-022-1694-1","volume":"38","author":"H Wang","year":"2022","unstructured":"Wang, H.: Continuity of the solution to the $$L_p$$ Minkowski problem in Gaussian probability space. Acta Math. Sin. (Engl. Ser.) 38, 2253\u20132264 (2022)","journal-title":"Acta Math. Sin. (Engl. Ser.)"},{"key":"740_CR47","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1007\/s11401-024-0011-y","volume":"45","author":"H Wang","year":"2024","unstructured":"Wang, H.: On existence of the even $$L_p$$ Gaussian Minkowski problem for $$p>n$$. Chinese Ann. Math. Ser. B 45, 179\u2013192 (2024)","journal-title":"Chinese Ann. Math. Ser. B"},{"key":"740_CR48","doi-asserted-by":"publisher","first-page":"1795","DOI":"10.1090\/tran\/7350","volume":"371","author":"Y Wu","year":"2019","unstructured":"Wu, Y., Xi, D., Leng, G.: On the discrete Orlicz Minkowski problem. Trans. Amer. Math. Soc. 371, 1795\u20131814 (2019)","journal-title":"Trans. Amer. Math. Soc."},{"key":"740_CR49","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1007\/s10711-019-00471-z","volume":"205","author":"Y Wu","year":"2020","unstructured":"Wu, Y., Xi, D., Leng, G.: On the discrete Orlicz Minkowski problem II. Geom. Dedicata. 205, 177\u2013190 (2020)","journal-title":"Geom. Dedicata."},{"key":"740_CR50","doi-asserted-by":"publisher","first-page":"350","DOI":"10.1016\/j.aim.2014.02.036","volume":"264","author":"D Xi","year":"2014","unstructured":"Xi, D., Jin, H., Leng, G.: The Orlicz Brunn-Minkowski inequality. Adv. Math. 264, 350\u2013374 (2014)","journal-title":"Adv. Math."},{"key":"740_CR51","doi-asserted-by":"publisher","first-page":"22","DOI":"10.1515\/ans-2022-0041","volume":"23","author":"D Xi","year":"2023","unstructured":"Xi, D., Yang, D., Zhang, G., Zhao, Y.: The $$L_{p}$$ chord Minkowski problem. Adv. Nonlinear Stud. 23, 22 (2023)","journal-title":"Adv. Nonlinear Stud."},{"key":"740_CR52","doi-asserted-by":"publisher","first-page":"4433","DOI":"10.1090\/proc\/14722","volume":"150","author":"F Xie","year":"2022","unstructured":"Xie, F.: The Orlicz Minkowski problem for general measures. Proc. Am. Math. Soc. 150, 4433\u20134445 (2022)","journal-title":"Proc. Am. Math. Soc."}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-025-00740-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-025-00740-7","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-025-00740-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,30]],"date-time":"2026-05-30T19:02:15Z","timestamp":1780167735000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-025-00740-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,4]]},"references-count":52,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,6]]}},"alternative-id":["740"],"URL":"https:\/\/doi.org\/10.1007\/s00454-025-00740-7","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,4]]},"assertion":[{"value":"12 January 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 April 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 May 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 July 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}