{"abstract":[{"text":"A tight frame is the orthogonal projection of some orthonormal basis of Rn onto Rk. We show that a set of vectors is a tight frame if and only if the set of all cross products of these vectors is a tight frame. We reformulate a range of problems on the volume of projections (or sections) of regular polytopes in terms of tight frames and write a first-order necessary condition for local extrema of these problems. As applications, we prove new results for the problem of maximization of the volume of zonotopes.","lang":"eng"}],"isi":1,"keyword":["General Mathematics","Tight frame","Grassmannian","zonotope"],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1804.10055"}],"day":"18","publisher":"Canadian Mathematical Society","type":"journal_article","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","intvolume":"        64","oa":1,"title":"Tight frames and related geometric problems","article_type":"original","quality_controlled":"1","external_id":{"isi":["000730165300021"],"arxiv":["1804.10055"]},"date_published":"2021-12-18T00:00:00Z","article_processing_charge":"No","doi":"10.4153/s000843952000096x","citation":{"chicago":"Ivanov, Grigory. “Tight Frames and Related Geometric Problems.” <i>Canadian Mathematical Bulletin</i>. Canadian Mathematical Society, 2021. <a href=\"https://doi.org/10.4153/s000843952000096x\">https://doi.org/10.4153/s000843952000096x</a>.","ama":"Ivanov G. Tight frames and related geometric problems. <i>Canadian Mathematical Bulletin</i>. 2021;64(4):942-963. doi:<a href=\"https://doi.org/10.4153/s000843952000096x\">10.4153/s000843952000096x</a>","ieee":"G. Ivanov, “Tight frames and related geometric problems,” <i>Canadian Mathematical Bulletin</i>, vol. 64, no. 4. Canadian Mathematical Society, pp. 942–963, 2021.","short":"G. Ivanov, Canadian Mathematical Bulletin 64 (2021) 942–963.","ista":"Ivanov G. 2021. Tight frames and related geometric problems. Canadian Mathematical Bulletin. 64(4), 942–963.","mla":"Ivanov, Grigory. “Tight Frames and Related Geometric Problems.” <i>Canadian Mathematical Bulletin</i>, vol. 64, no. 4, Canadian Mathematical Society, 2021, pp. 942–63, doi:<a href=\"https://doi.org/10.4153/s000843952000096x\">10.4153/s000843952000096x</a>.","apa":"Ivanov, G. (2021). Tight frames and related geometric problems. <i>Canadian Mathematical Bulletin</i>. Canadian Mathematical Society. <a href=\"https://doi.org/10.4153/s000843952000096x\">https://doi.org/10.4153/s000843952000096x</a>"},"status":"public","_id":"10860","page":"942-963","acknowledgement":"The author was supported by the Swiss National Science Foundation grant 200021_179133. The author acknowledges the financial support from the Ministry of Education and Science of the Russian Federation in the framework of MegaGrant no. 075-15-2019-1926.","author":[{"last_name":"Ivanov","full_name":"Ivanov, Grigory","first_name":"Grigory","id":"87744F66-5C6F-11EA-AFE0-D16B3DDC885E"}],"publication":"Canadian Mathematical Bulletin","department":[{"_id":"UlWa"}],"date_created":"2022-03-18T09:55:59Z","date_updated":"2023-09-05T12:43:09Z","publication_identifier":{"eissn":["1496-4287"],"issn":["0008-4395"]},"volume":64,"year":"2021","language":[{"iso":"eng"}],"issue":"4","scopus_import":"1","oa_version":"Preprint","publication_status":"published","month":"12"}