{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T08:11:00Z","timestamp":1767859860293,"version":"3.49.0"},"reference-count":22,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2019,8,19]],"date-time":"2019-08-19T00:00:00Z","timestamp":1566172800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,8,19]],"date-time":"2019-08-19T00:00:00Z","timestamp":1566172800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"publisher","award":["1440140"],"award-info":[{"award-number":["1440140"]}],"id":[{"id":"10.13039\/100000086","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":[[2020,1]]},"DOI":"10.1007\/s00454-019-00115-9","type":"journal-article","created":{"date-parts":[[2019,8,19]],"date-time":"2019-08-19T17:02:43Z","timestamp":1566234163000},"page":"209-228","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Cube is a Strict Local Maximizer for the Illumination Number"],"prefix":"10.1007","volume":"63","author":[{"given":"Galyna","family":"Livshyts","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Konstantin","family":"Tikhomirov","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,8,19]]},"reference":[{"key":"115_CR1","volume-title":"Classical Topics in Discrete Geometry. CMS Books in Mathematics\/Ouvrages de Math\u00e9matiques de la SMC","author":"K Bezdek","year":"2010","unstructured":"Bezdek, K.: Classical Topics in Discrete Geometry. CMS Books in Mathematics\/Ouvrages de Math\u00e9matiques de la SMC. Springer, New York (2010)"},{"issue":"2","key":"115_CR2","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1007\/s00454-011-9369-1","volume":"47","author":"K Bezdek","year":"2012","unstructured":"Bezdek, K.: Illuminating spindle convex bodies and minimizing the volume of spherical sets of constant width. Discrete Comput. Geom. 47(2), 275\u2013287 (2012)","journal-title":"Discrete Comput. Geom."},{"issue":"1","key":"115_CR3","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1023\/A:1004965512469","volume":"68","author":"K Bezdek","year":"1997","unstructured":"Bezdek, K., Bisztriczky, T.: A proof of Hadwiger\u2019s covering conjecture for dual cyclic polytopes. Geom. Dedicata 68(1), 29\u201341 (1997)","journal-title":"Geom. Dedicata"},{"key":"115_CR4","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/978-3-319-78434-2_1","volume-title":"Discrete Geometry and Symmetry. Springer Proceedings in Mathematics & Statistics","author":"K Bezdek","year":"2018","unstructured":"Bezdek, K., Khan, M.A.: The geometry of homothetic covering and illumination. In: Conder, M.D.E., Deza, A., Ivi\u0107 Weiss, A. (eds.) Discrete Geometry and Symmetry. Springer Proceedings in Mathematics & Statistics, vol. 234, pp. 1\u201330. Springer, Cham (2018)"},{"issue":"3\u20134","key":"115_CR5","first-page":"1029","volume":"58","author":"VG Boltyanski\u012d","year":"1996","unstructured":"Boltyanski\u012d, V.G.: Solution of the illumination problem for belt-bodies. Math. Notes 58(3\u20134), 1029\u20131032 (1996)","journal-title":"Math. Notes"},{"key":"115_CR6","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-59237-9","volume-title":"Excursions into Combinatorial Geometry. Universitext.","author":"V Boltyanski","year":"1997","unstructured":"Boltyanski, V., Martini, H., Soltan, P.S.: Excursions into Combinatorial Geometry. Universitext. Springer, Berlin (1997)"},{"issue":"2","key":"115_CR7","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1007\/BF01388911","volume":"88","author":"J Bourgain","year":"1987","unstructured":"Bourgain, J., Milman, V.D.: New volume ratio properties for convex symmetric bodies in $${ R}^n$$. Invent. Math. 88(2), 319\u2013340 (1987)","journal-title":"Invent. Math."},{"key":"115_CR8","first-page":"121","volume":"12","author":"H Hadwiger","year":"1957","unstructured":"Hadwiger, H.: Ungel\u00f6ste probleme, Nr.\u00a020. Elem. Math. 12, 121 (1957)","journal-title":"Elem. Math."},{"key":"115_CR9","first-page":"130","volume":"15","author":"H Hadwiger","year":"1960","unstructured":"Hadwiger, H.: Ungel\u00f6ste probleme. Nr.\u00a038. Elem. Math. 15, 130\u2013131 (1960)","journal-title":"Elem. Math."},{"key":"115_CR10","unstructured":"Iriyeh, H., Shibata, M.: Symmetric Mahler\u2019s conjecture for the volume product in the three dimensional case (2017). \narXiv:1706.01749"},{"issue":"3","key":"115_CR11","doi-asserted-by":"publisher","first-page":"870","DOI":"10.1007\/s00039-008-0669-4","volume":"18","author":"G Kuperberg","year":"2008","unstructured":"Kuperberg, G.: From the Mahler conjecture to Gauss linking integrals. Geom. Funct. Anal. 18(3), 870\u2013892 (2008)","journal-title":"Geom. Funct. Anal."},{"issue":"3","key":"115_CR12","doi-asserted-by":"publisher","first-page":"501","DOI":"10.1112\/jlms\/s2-30.3.501","volume":"30","author":"M Lassak","year":"1984","unstructured":"Lassak, M.: Solution of Hadwiger\u2019s covering problem for centrally symmetric convex bodies in $$E^3$$. J. Lond. Math. Soc. II 30(3), 501\u2013511 (1984)","journal-title":"J. Lond. Math. Soc. II"},{"issue":"5","key":"115_CR13","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1007\/BF01900507","volume":"6","author":"FW Levi","year":"1955","unstructured":"Levi, F.W.: \u00dcberdeckung eines Eibereiches durch Parallelverschiebungen seines offenen Kerns. Arch. Math. 6(5), 369\u2013370 (1955)","journal-title":"Arch. Math."},{"key":"115_CR14","unstructured":"Livshyts, G., Tikhomirov, K.: Randomized coverings of a convex body with its homothetic copies, and illumination (2016). \narXiv:1606.08876"},{"key":"115_CR15","first-page":"383","volume-title":"Intuitive Geometry (Si\u00f3fok, 1985). Colloquia Mathematica Societatis J\u00e1nos Bolyai","author":"H Martini","year":"1987","unstructured":"Martini, H.: Some results and problems around zonotopes. In: B\u00f6r\u00f6czky, K., Fejes T\u00f3th, G. (eds.) Intuitive Geometry (Si\u00f3fok, 1985). Colloquia Mathematica Societatis J\u00e1nos Bolyai, vol. 48, pp. 383\u2013418. North-Holland, Amsterdam (1987)"},{"issue":"2","key":"115_CR16","doi-asserted-by":"publisher","first-page":"630","DOI":"10.1112\/S0025579315000406","volume":"62","author":"M Nasz\u00f3di","year":"2016","unstructured":"Nasz\u00f3di, M.: A spiky ball. Mathematika 62(2), 630\u2013636 (2016)","journal-title":"Mathematika"},{"issue":"3","key":"115_CR17","doi-asserted-by":"publisher","first-page":"419","DOI":"10.1215\/00127094-2010-042","volume":"154","author":"F Nazarov","year":"2010","unstructured":"Nazarov, F., Petrov, F., Ryabogin, D., Zvavitch, A.: A remark on the Mahler conjecture: local minimality of the unit cube. Duke Math. J. 154(3), 419\u2013430 (2010)","journal-title":"Duke Math. J."},{"key":"115_CR18","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1112\/S0025579300001030","volume":"4","author":"CA Rogers","year":"1957","unstructured":"Rogers, C.A.: A note on coverings. Mathematika 4, 1\u20136 (1957)","journal-title":"Mathematika"},{"key":"115_CR19","doi-asserted-by":"publisher","first-page":"220","DOI":"10.1007\/BF01899997","volume":"8","author":"CA Rogers","year":"1957","unstructured":"Rogers, C.A., Shephard, G.C.: The difference body of a convex body. Arch. Math. 8, 220\u2013233 (1957)","journal-title":"Arch. Math."},{"issue":"2","key":"115_CR20","doi-asserted-by":"publisher","first-page":"180","DOI":"10.1112\/S0025579300015175","volume":"35","author":"O Schramm","year":"1988","unstructured":"Schramm, O.: Illuminating sets of constant width. Mathematika 35(2), 180\u2013189 (1988)","journal-title":"Mathematika"},{"issue":"2","key":"115_CR21","doi-asserted-by":"publisher","first-page":"372","DOI":"10.1112\/S0025579316000292","volume":"63","author":"K Tikhomirov","year":"2017","unstructured":"Tikhomirov, K.: Illumination of convex bodies with many symmetries. Mathematika 63(2), 372\u2013382 (2017)","journal-title":"Mathematika"},{"issue":"9","key":"115_CR22","doi-asserted-by":"publisher","first-page":"2551","DOI":"10.1007\/s11425-010-4087-3","volume":"53","author":"C Zong","year":"2010","unstructured":"Zong, C.: A quantitative program for Hadwigers covering conjecture. Sci. China Math. 53(9), 2551\u20132560 (2010)","journal-title":"Sci. China Math."}],"container-title":["Discrete & Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-019-00115-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00454-019-00115-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-019-00115-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,8,17]],"date-time":"2020-08-17T23:25:08Z","timestamp":1597706708000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00454-019-00115-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8,19]]},"references-count":22,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2020,1]]}},"alternative-id":["115"],"URL":"https:\/\/doi.org\/10.1007\/s00454-019-00115-9","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,8,19]]},"assertion":[{"value":"24 December 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 May 2019","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 July 2019","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 August 2019","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}