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<mml:mi>v<\/mml:mi>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are unit vectors in\u00a0<jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb {C}}^d$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>C<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mi>d<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and <jats:inline-formula><jats:alternatives><jats:tex-math>$$t_1,\\dots ,t_n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>t<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mo>\u22ef<\/mml:mo>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>t<\/mml:mi>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are non-negative numbers satisfying <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\sum _{k=1}^nt_k^2 = 1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msubsup>\n                      <mml:mo>\u2211<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mi>k<\/mml:mi>\n                        <mml:mo>=<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:mrow>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msubsup>\n                    <mml:msubsup>\n                      <mml:mi>t<\/mml:mi>\n                      <mml:mi>k<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msubsup>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, then there exists a unit vector <jats:italic>v<\/jats:italic> in <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb {C}}^d$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>C<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mi>d<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for which <jats:inline-formula><jats:alternatives><jats:tex-math>$$|\\langle v_k,v \\rangle | \\ge t_k$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>\u27e8<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>v<\/mml:mi>\n                        <mml:mi>k<\/mml:mi>\n                      <\/mml:msub>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>v<\/mml:mi>\n                      <mml:mo>\u27e9<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                      <mml:mo>\u2265<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:msub>\n                      <mml:mi>t<\/mml:mi>\n                      <mml:mi>k<\/mml:mi>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for every\u00a0<jats:italic>k<\/jats:italic>. Here we present a streamlined version of Ball\u2019s original proof.<\/jats:p>","DOI":"10.1007\/s00454-022-00423-7","type":"journal-article","created":{"date-parts":[[2022,8,27]],"date-time":"2022-08-27T19:02:27Z","timestamp":1661626947000},"page":"683-687","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The Complex Plank Problem, Revisited"],"prefix":"10.1007","volume":"71","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6819-5121","authenticated-orcid":false,"given":"Oscar","family":"Ortega-Moreno","sequence":"first","affiliation":[],"role":[{"role":"author","vocab":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,8,27]]},"reference":[{"key":"423_CR1","unstructured":"Ambrus, G.: Analytic and Probabilistic Problems in Discrete Geometry. 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