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In particular, we show the optimality of regular simplexes, describe some $M$-sets of small cardinality, and define and discuss spherical $f$-designs.<\/jats:p>","DOI":"10.1137\/20m1324831","type":"journal-article","created":{"date-parts":[[2021,7,12]],"date-time":"2021-07-12T11:54:33Z","timestamp":1626090873000},"page":"1578-1591","source":"Crossref","is-referenced-by-count":1,"title":["Majorization and Minimal Energy on Spheres"],"prefix":"10.1137","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5946-4841","authenticated-orcid":true,"given":"Oleg R.","family":"Musin","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,7,12]]},"reference":[{"key":"atypb1","first-page":"19","volume":"107","author":"Andreev N. N.","year":"1999","journal-title":"Math. 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