{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:46:47Z","timestamp":1740109607145,"version":"3.37.3"},"reference-count":18,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,3,30]],"date-time":"2024-03-30T00:00:00Z","timestamp":1711756800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,3,30]],"date-time":"2024-03-30T00:00:00Z","timestamp":1711756800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["2261055"],"award-info":[{"award-number":["2261055"]}],"id":[{"id":"10.13039\/501100000266","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":[[2025,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>K<\/jats:italic> be a convex body (a compact convex set) in <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {R}^d$$<\/jats:tex-math><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:mi>d<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, that contains a copy of another body <jats:italic>S<\/jats:italic> in every possible orientation. Is it always possible to continuously move any one copy of <jats:italic>S<\/jats:italic> into another, inside <jats:italic>K<\/jats:italic>? As a stronger question, is it always possible to continuously select, for each orientation, one copy of <jats:italic>S<\/jats:italic> in that orientation? These questions were asked by Croft. We show that, in two dimensions, the stronger question always has an affirmative answer. We also show that in three dimensions the answer is negative, even for the case when <jats:italic>S<\/jats:italic> is a line segment \u2013 but that in any dimension the first question has a positive answer when <jats:italic>S<\/jats:italic> is a line segment. And we prove that, surprisingly, the answer to the first question is negative in dimensions four and higher for general <jats:italic>S<\/jats:italic>.<\/jats:p>","DOI":"10.1007\/s00454-024-00639-9","type":"journal-article","created":{"date-parts":[[2024,3,30]],"date-time":"2024-03-30T22:01:16Z","timestamp":1711836076000},"page":"149-176","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Rotation Inside Convex Kakeya Sets"],"prefix":"10.1007","volume":"73","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9904-7188","authenticated-orcid":false,"given":"Barnab\u00e1s","family":"Janzer","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,3,30]]},"reference":[{"issue":"1","key":"639_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2140\/pjm.1978.77.1","volume":"77","author":"D Amir","year":"1978","unstructured":"Amir, D.: Chebyshev centers and uniform convexity. Pac. J. Math. 77(1), 1\u20136 (1978)","journal-title":"Pac. J. Math."},{"key":"639_CR2","doi-asserted-by":"crossref","unstructured":"Amir, D.: Best simultaneous approximation (Chebyshev centers). In: Parametric Optimization and Approximation, pp. 19\u201335. Springer, Berlin (1984)","DOI":"10.1007\/978-3-0348-6253-0_2"},{"issue":"1","key":"639_CR3","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1515\/advgeom-2020-0030","volume":"21","author":"SW Bae","year":"2021","unstructured":"Bae, S.W., Cabello, S., Cheong, O., Choi, Y., Stehn, F., Yoon, S.D.: The reverse Kakeya problem. Adv. Geom. 21(1), 75\u201384 (2021)","journal-title":"Adv. Geom."},{"key":"639_CR4","unstructured":"Besicovitch, A.: Sur deux questions de l\u2019integrabilit\u00e9. J. Phys. Math. Soc. State Perm Univ. 2 (1920)"},{"issue":"1","key":"639_CR5","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1007\/BF01171101","volume":"27","author":"A Besicovitch","year":"1928","unstructured":"Besicovitch, A.: On Kakeya\u2019s problem and a similar one. Mathematische Zeitschrift 27(1), 312\u2013320 (1928)","journal-title":"Mathematische Zeitschrift"},{"issue":"2","key":"639_CR6","doi-asserted-by":"publisher","first-page":"256","DOI":"10.1007\/s000390050087","volume":"9","author":"J Bourgain","year":"1999","unstructured":"Bourgain, J.: On the dimension of Kakeya sets and related maximal inequalities. Geom. Funct. Anal. GAFA 9(2), 256\u2013282 (1999)","journal-title":"Geom. Funct. Anal. GAFA"},{"issue":"1","key":"639_CR7","doi-asserted-by":"publisher","first-page":"184","DOI":"10.1112\/S0025579316000188","volume":"63","author":"M Cs\u00f6rnyei","year":"2017","unstructured":"Cs\u00f6rnyei, M., H\u00e9ra, K., Laczkovich, M.: Closed sets with the Kakeya property. Mathematika 63(1), 184\u2013195 (2017)","journal-title":"Mathematika"},{"issue":"2","key":"639_CR8","doi-asserted-by":"publisher","first-page":"114","DOI":"10.1080\/00029890.1971.11992708","volume":"78","author":"F Cunningham Jr","year":"1971","unstructured":"Cunningham, F., Jr.: The Kakeya problem for simply connected and for star-shaped sets. Am. Math. Mon. 78(2), 114\u2013129 (1971)","journal-title":"Am. Math. Mon."},{"issue":"3","key":"639_CR9","doi-asserted-by":"publisher","first-page":"417","DOI":"10.1017\/S0305004100046867","volume":"69","author":"RO Davies","year":"1971","unstructured":"Davies, R.O.: Some remarks on the Kakeya problem. Math. Proc. Camb. Philos. Soc. 69(3), 417\u2013421 (1971)","journal-title":"Math. Proc. Camb. Philos. Soc."},{"issue":"4","key":"639_CR10","doi-asserted-by":"publisher","first-page":"1093","DOI":"10.1090\/S0894-0347-08-00607-3","volume":"22","author":"Z Dvir","year":"2009","unstructured":"Dvir, Z.: On the size of Kakeya sets in finite fields. J. Am. Math. Soc. 22(4), 1093\u20131097 (2009)","journal-title":"J. Am. Math. Soc."},{"issue":"2","key":"639_CR11","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1017\/S0305004100056681","volume":"87","author":"KJ Falconer","year":"1980","unstructured":"Falconer, K.J.: Continuity properties of k-plane integrals and Besicovitch sets. Math. Proc. Camb. Philos. Soc. 87(2), 221\u2013226 (1980)","journal-title":"Math. Proc. Camb. Philos. Soc."},{"issue":"2","key":"639_CR12","doi-asserted-by":"publisher","first-page":"330","DOI":"10.2307\/1970864","volume":"94","author":"C Fefferman","year":"1971","unstructured":"Fefferman, C.: The multiplier problem for the ball. Ann. Math. 94(2), 330\u2013336 (1971)","journal-title":"Ann. Math."},{"key":"639_CR13","doi-asserted-by":"crossref","unstructured":"H\u00e9ra, K., Laczkovich, M.: The Kakeya problem for circular arcs. Acta Mathematica Hungarica (2016)","DOI":"10.1007\/s10474-016-0663-5"},{"issue":"2","key":"639_CR14","doi-asserted-by":"publisher","first-page":"411","DOI":"10.5186\/aasfm.2011.3639","volume":"36","author":"E J\u00e4rvenp\u00e4\u00e4","year":"2011","unstructured":"J\u00e4rvenp\u00e4\u00e4, E., J\u00e4rvenp\u00e4\u00e4, M., Keleti, T., M\u00e1th\u00e9, A.: Continuously parametrized Besicovitch sets in $${ R}^n$$. Annales Academi\u00e6Scientiarum Fennic\u00e6. Mathematica 36(2), 411\u2013421 (2011)","journal-title":"Annales Academi\u00e6Scientiarum Fennic\u00e6. Mathematica"},{"issue":"1","key":"639_CR15","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1007\/BF02868476","volume":"87","author":"N Katz","year":"2002","unstructured":"Katz, N., Tao, T.: New bounds for Kakeya problems. Journal d\u2019Analyse Math\u00e9matique 87(1), 231\u2013263 (2002)","journal-title":"Journal d\u2019Analyse Math\u00e9matique"},{"issue":"1","key":"639_CR16","doi-asserted-by":"publisher","first-page":"111","DOI":"10.2140\/pjm.1999.190.111","volume":"190","author":"L Kolasa","year":"1999","unstructured":"Kolasa, L., Wolff, T.: On some variants of the Kakeya problem. Pac. J. Math. 190(1), 111\u2013154 (1999)","journal-title":"Pac. J. Math."},{"issue":"3","key":"639_CR17","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1007\/BF01458387","volume":"83","author":"J P\u00e1l","year":"1921","unstructured":"P\u00e1l, J.: Ein minimumproblem f\u00fcr ovale. Mathematische Annalen 83(3), 311\u2013319 (1921)","journal-title":"Mathematische Annalen"},{"key":"639_CR18","first-page":"144","volume":"10","author":"HJ van Alphen","year":"1942","unstructured":"van Alphen, H.J.: Uitbreiding van een stelling von Besicovitch. Mathematica (Zutphen) B 10, 144\u2013157 (1942)","journal-title":"Mathematica (Zutphen) B"}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-024-00639-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-024-00639-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-024-00639-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,1,6]],"date-time":"2025-01-06T16:04:46Z","timestamp":1736179486000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-024-00639-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,30]]},"references-count":18,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,1]]}},"alternative-id":["639"],"URL":"https:\/\/doi.org\/10.1007\/s00454-024-00639-9","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"type":"print","value":"0179-5376"},{"type":"electronic","value":"1432-0444"}],"subject":[],"published":{"date-parts":[[2024,3,30]]},"assertion":[{"value":"2 December 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 February 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 February 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 March 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}