{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T00:28:48Z","timestamp":1767140928315,"version":"build-2238731810"},"reference-count":10,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2021,1,22]],"date-time":"2021-01-22T00:00:00Z","timestamp":1611273600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,1,22]],"date-time":"2021-01-22T00:00:00Z","timestamp":1611273600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100003443","name":"Ministry of Education and Science of the Russian Federation","doi-asserted-by":"crossref","award":["MegaGrant no. 075-15-2019-1926"],"award-info":[{"award-number":["MegaGrant no. 075-15-2019-1926"]}],"id":[{"id":"10.13039\/501100003443","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100012550","name":"Nemzeti Kutat\u00e1si, Fejleszt\u00e9si \u00e9s Innovaci\u00f3s Alap","doi-asserted-by":"publisher","award":["NKFIH Grant K119670"],"award-info":[{"award-number":["NKFIH Grant K119670"]}],"id":[{"id":"10.13039\/501100012550","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000608","name":"London Mathematical Society","doi-asserted-by":"publisher","award":["ECF-1920-69"],"award-info":[{"award-number":["ECF-1920-69"]}],"id":[{"id":"10.13039\/501100000608","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":[[2022,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    For fixed\n                    <jats:italic>k<\/jats:italic>\n                    we prove exponential lower bounds on the equilateral number of subspaces of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\ell _{\\infty }^n$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msubsup>\n                            <mml:mi>\u2113<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u221e<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    of codimension\u00a0\n                    <jats:italic>k<\/jats:italic>\n                    . In particular, we show that subspaces of codimension\u00a02 of\u00a0\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\ell _{\\infty }^{n+2}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msubsup>\n                            <mml:mi>\u2113<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u221e<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and subspaces of codimension\u00a03 of\u00a0\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\ell _{\\infty }^{n+3}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msubsup>\n                            <mml:mi>\u2113<\/mml:mi>\n                            <mml:mrow>\n                              <mml:mi>\u221e<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mn>3<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msubsup>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    have an equilateral set of cardinality\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$n+1$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    if\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$n\\ge 7$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\u2265<\/mml:mo>\n                            <mml:mn>7<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$n\\ge 12$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\u2265<\/mml:mo>\n                            <mml:mn>12<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    respectively. Moreover, the same is true for every normed space of dimension\u00a0\n                    <jats:italic>n<\/jats:italic>\n                    , whose unit ball is a centrally symmetric polytope with at most\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$${4n}\/{3}-o(n)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mn>4<\/mml:mn>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>\/<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                            <mml:mo>-<\/mml:mo>\n                            <mml:mi>o<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    pairs of facets.\n                  <\/jats:p>","DOI":"10.1007\/s00454-020-00272-2","type":"journal-article","created":{"date-parts":[[2021,1,22]],"date-time":"2021-01-22T10:27:47Z","timestamp":1611311267000},"page":"882-893","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Large Equilateral Sets in Subspaces of $$\\ell _\\infty ^n$$ of Small Codimension"],"prefix":"10.1007","volume":"67","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4939-4835","authenticated-orcid":false,"given":"N\u00f3ra","family":"Frankl","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,1,22]]},"reference":[{"key":"272_CR1","unstructured":"Ball, K.: An elementary introduction to modern convex geometry. In: Flavors of Geometry. Math. Sci. Res. Inst. Publ., vol. 31, pp. 1\u201358. Cambridge Univ. Press, Cambridge (1997)"},{"issue":"2","key":"272_CR2","first-page":"303","volume":"40","author":"P Bra\u00df","year":"1999","unstructured":"Bra\u00df, P.: On equilateral simplices in normed spaces. Beitr\u00e4ge Algebra Geom. 40(2), 303\u2013307 (1999)","journal-title":"Beitr\u00e4ge Algebra Geom."},{"issue":"4","key":"272_CR3","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1023\/A:1006727810727","volume":"86","author":"BV Dekster","year":"2000","unstructured":"Dekster, B.V.: Simplexes with prescribed edge lengths in Minkowski and Banach spaces. Acta Math. Hungar. 86(4), 343\u2013358 (2000)","journal-title":"Acta Math. Hungar."},{"key":"272_CR4","doi-asserted-by":"crossref","unstructured":"Gr\u00fcnbaum, B.: Convex Polytopes. Graduate Texts in Mathematics, vol. 221. Springer, New York (2003)","DOI":"10.1007\/978-1-4613-0019-9"},{"issue":"10","key":"272_CR5","doi-asserted-by":"publisher","first-page":"1340","DOI":"10.1080\/01630563.2014.930482","volume":"35","author":"T Kobos","year":"2014","unstructured":"Kobos, T.: Equilateral dimension of certain classes of normed spaces. Numer. Funct. Anal. Optim. 35(10), 1340\u20131358 (2014)","journal-title":"Numer. Funct. Anal. Optim."},{"key":"272_CR6","first-page":"88","volume":"329","author":"VV Makeev","year":"2005","unstructured":"Makeev, V.V.: On equilateral simplices in a four-dimensional normed space. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 329, 88\u201391 (2005). (in Russian)","journal-title":"Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI)"},{"key":"272_CR7","doi-asserted-by":"publisher","first-page":"369","DOI":"10.1090\/S0002-9939-1971-0275294-8","volume":"29","author":"CM Petty","year":"1971","unstructured":"Petty, C.M.: Equilateral sets in Minkowski spaces. Proc. Am. Math. Soc. 29, 369\u2013374 (1971)","journal-title":"Proc. Am. Math. Soc."},{"issue":"6","key":"272_CR8","first-page":"1303","volume":"222","author":"PS Soltan","year":"1975","unstructured":"Soltan, P.S.: Analogues of regular simplexes in normed spaces. Dokl. Akad. Nauk SSSR 222(6), 1303\u20131305 (1975). (in Russian)","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"272_CR9","doi-asserted-by":"crossref","unstructured":"Swanepoel, K.J.: Combinatorial distance geometry in normed spaces. In: New Trends in Intuitive Geometry. Bolyai Soc. Math. Stud., vol. 27, pp. 407\u2013458. J\u00e1nos Bolyai Math. Soc., Budapest (2018)","DOI":"10.1007\/978-3-662-57413-3_17"},{"issue":"1","key":"272_CR10","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1090\/S0002-9939-07-08916-2","volume":"136","author":"KJ Swanepoel","year":"2008","unstructured":"Swanepoel, K.J., Villa, R.: A lower bound for the equilateral number of normed spaces. Proc. Am. Math. Soc. 136(1), 127\u2013131 (2008)","journal-title":"Proc. Am. Math. Soc."}],"updated-by":[{"DOI":"10.1007\/s00454-021-00293-5","type":"correction","label":"Correction","source":"publisher","updated":{"date-parts":[[2021,3,19]],"date-time":"2021-03-19T00:00:00Z","timestamp":1616112000000}}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-020-00272-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-020-00272-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-020-00272-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,3,11]],"date-time":"2022-03-11T11:33:38Z","timestamp":1646998418000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-020-00272-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,22]]},"references-count":10,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,4]]}},"alternative-id":["272"],"URL":"https:\/\/doi.org\/10.1007\/s00454-020-00272-2","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,22]]},"assertion":[{"value":"8 May 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 November 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 December 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 January 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 March 2021","order":5,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Correction","order":6,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"A Correction to this paper has been published:","order":7,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"https:\/\/doi.org\/10.1007\/s00454-021-00293-5","URL":"https:\/\/doi.org\/10.1007\/s00454-021-00293-5","order":8,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}}]}}