{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:46:10Z","timestamp":1740109570225,"version":"3.37.3"},"reference-count":13,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2021,5,10]],"date-time":"2021-05-10T00:00:00Z","timestamp":1620604800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,5,10]],"date-time":"2021-05-10T00:00:00Z","timestamp":1620604800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004895","name":"European Social Fund","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004895","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":[[2021,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We give a full classification of vertex-transitive zonotopes. We prove that a vertex-transitive zonotope is a <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Gamma $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u0393<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-permutahedron for some finite reflection group <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Gamma \\subset {{\\,\\mathrm{O}\\,}}(\\mathbb {R}^d)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u0393<\/mml:mi>\n                    <mml:mo>\u2282<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mspace\/>\n                      <mml:mi>O<\/mml:mi>\n                      <mml:mspace\/>\n                    <\/mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mi>R<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mi>d<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The same holds true for zonotopes in which all vertices are on a common sphere, and all edges are of the same length. The classification of these then follows from the classification of finite reflection groups. We prove that root systems can be characterized as those centrally symmetric sets of vectors, for which all intersections with half-spaces, that contain exactly half the vectors, are congruent. We provide a further sufficient condition for a centrally symmetric set to be a root system.<\/jats:p>","DOI":"10.1007\/s00454-021-00303-6","type":"journal-article","created":{"date-parts":[[2021,5,10]],"date-time":"2021-05-10T14:04:17Z","timestamp":1620655457000},"page":"1446-1462","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Classification of Vertex-Transitive Zonotopes"],"prefix":"10.1007","volume":"66","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3817-9494","authenticated-orcid":false,"given":"Martin","family":"Winter","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,5,10]]},"reference":[{"issue":"3","key":"303_CR1","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1007\/BF02429904","volume":"6","author":"L Babai","year":"1977","unstructured":"Babai, L.: Symmetry groups of vertex-transitive polytopes. Geometriae Dedicata 6(3), 331\u2013337 (1977)","journal-title":"Geometriae Dedicata"},{"key":"303_CR2","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1112\/plms\/s2-38.1.327","volume":"38","author":"HSM Coxeter","year":"1935","unstructured":"Coxeter, H.S.M.: Wythoff\u2019s construction for uniform polytopes. Proc. Lond. Math. Soc. 38, 327\u2013339 (1935)","journal-title":"Proc. Lond. Math. Soc."},{"issue":"916","key":"303_CR3","doi-asserted-by":"publisher","first-page":"401","DOI":"10.1098\/rsta.1954.0003","volume":"246","author":"HSM Coxeter","year":"1954","unstructured":"Coxeter, H.S.M., Longuet-Higgins, M.S., Miller, J.C.P.: Uniform polyhedra. Philos. Trans. R. Soc. Lond. Ser. A 246(916), 401\u2013450 (1954)","journal-title":"Philos. Trans. R. Soc. Lond. Ser. A"},{"issue":"11","key":"303_CR4","doi-asserted-by":"publisher","first-page":"4143","DOI":"10.1016\/j.jalgebra.2009.09.031","volume":"322","author":"M Dutour Sikiri\u0107","year":"2009","unstructured":"Dutour Sikiri\u0107, M., Ellis, G.: Wythoff polytopes and low-dimensional homology of Mathieu groups. J. Algebra 322(11), 4143\u20134150 (2009)","journal-title":"J. Algebra"},{"issue":"4","key":"303_CR5","doi-asserted-by":"publisher","first-page":"891","DOI":"10.1007\/s13366-016-0286-6","volume":"57","author":"R Ehrenborg","year":"2016","unstructured":"Ehrenborg, R., Klivans, C., Reading, N.: Coxeter arrangements in three dimensions. Beitr. Algebra Geom. 57(4), 891\u2013897 (2016)","journal-title":"Beitr. Algebra Geom."},{"key":"303_CR6","doi-asserted-by":"publisher","first-page":"386","DOI":"10.1016\/j.aim.2015.10.021","volume":"288","author":"E Friese","year":"2016","unstructured":"Friese, E., Ladisch, F.: Affine symmetries of orbit polytopes. Adv. Math. 288, 386\u2013425 (2016)","journal-title":"Adv. Math."},{"key":"303_CR7","doi-asserted-by":"crossref","unstructured":"Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990)","DOI":"10.1017\/CBO9780511623646"},{"key":"303_CR8","unstructured":"Johnson, N.W.: The Theory of Uniform Polytopes and Honeycombs. PhD thesis, University of Toronto (1966)"},{"key":"303_CR9","doi-asserted-by":"publisher","DOI":"10.1017\/9781316216477","volume-title":"Geometries and Transformations","author":"NW Johnson","year":"2018","unstructured":"Johnson, N.W.: Geometries and Transformations. Cambridge University Press, Cambridge (2018)"},{"key":"303_CR10","doi-asserted-by":"crossref","unstructured":"Kane, R.: Reflection Groups and Invariant Theory. CMS Books in Mathematics, vol. 5. Springer, New York (2001)","DOI":"10.1007\/978-1-4757-3542-0"},{"issue":"3","key":"303_CR11","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1007\/s00454-011-9363-7","volume":"46","author":"CJ Klivans","year":"2011","unstructured":"Klivans, C.J., Swartz, E.: Projection volumes of hyperplane arrangements. Discrete Comput. Geom. 46(3), 417\u2013426 (2011)","journal-title":"Discrete Comput. Geom."},{"key":"303_CR12","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1090\/S0002-9947-1971-0279689-2","volume":"159","author":"P McMullen","year":"1971","unstructured":"McMullen, P.: On zonotopes. Trans. Am. Math. Soc. 159, 91\u2013109 (1971)","journal-title":"Trans. Am. Math. Soc."},{"key":"303_CR13","doi-asserted-by":"crossref","unstructured":"Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)","DOI":"10.1007\/978-1-4613-8431-1"}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-021-00303-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-021-00303-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-021-00303-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,31]],"date-time":"2021-10-31T05:04:36Z","timestamp":1635656676000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-021-00303-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,10]]},"references-count":13,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,12]]}},"alternative-id":["303"],"URL":"https:\/\/doi.org\/10.1007\/s00454-021-00303-6","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"type":"print","value":"0179-5376"},{"type":"electronic","value":"1432-0444"}],"subject":[],"published":{"date-parts":[[2021,5,10]]},"assertion":[{"value":"9 January 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 November 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 March 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 May 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}