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Specifically, we determine the Banach\u2013Mazur distance between the cube and its dual (the cross-polytope) in <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\mathbb {R}^3$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>R<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mn>3<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> and <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\mathbb {R}^4$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>R<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mn>4<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. In dimension three this distance is equal to <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\frac{9}{5}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mfrac>\n                    <mml:mn>9<\/mml:mn>\n                    <mml:mn>5<\/mml:mn>\n                  <\/mml:mfrac>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, and in dimension four, it is equal to 2. These findings confirm well-known conjectures, which were based on numerical data. Additionally, in dimension two, we use the asymmetry constant to provide a geometric construction of a family of convex bodies that are equidistant to all symmetric convex bodies.<\/jats:p>","DOI":"10.1007\/s00454-024-00641-1","type":"journal-article","created":{"date-parts":[[2024,4,13]],"date-time":"2024-04-13T16:10:05Z","timestamp":1713024605000},"page":"399-427","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On the Banach\u2013Mazur Distance in Small Dimensions"],"prefix":"10.1007","volume":"74","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1571-5549","authenticated-orcid":false,"given":"Tomasz","family":"Kobos","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marin","family":"Varivoda","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,4,13]]},"reference":[{"key":"641_CR1","unstructured":"Banach, S.: Th\u00e9orie des op\u00e9rations lin\u00e9aires. 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