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With the absence of a least element for dimensions <jats:inline-formula><jats:alternatives><jats:tex-math>$$d\\ge 3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>d<\/mml:mi><mml:mo>\u2265<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the set of all minimal elements in the collection of all copulas turns out to be a natural and quite important extreme negative dependence concept. We investigate several sufficient conditions, and we provide a necessary condition for a copula to be minimal. The sufficient conditions are related to the extreme negative dependence concept of <jats:italic>d<\/jats:italic>-countermonotonicity and the necessary condition is related to the collection of all copulas minimizing multivariate Kendall\u2019s tau. The concept of minimal copulas has already been proved to be useful in various continuous and concordance order preserving optimization problems including variance minimization and the detection of lower bounds for certain measures of concordance. We substantiate this key role of minimal copulas by showing that every continuous and concordance order preserving functional on copulas is minimized by some minimal copula, and, in the case the continuous functional is even strictly concordance order preserving, it is minimized by minimal copulas only. Applying the above results, we may conclude that every minimizer of Spearman\u2019s rho is also a minimizer of Kendall\u2019s tau.<\/jats:p>","DOI":"10.1007\/s10957-019-01618-4","type":"journal-article","created":{"date-parts":[[2019,12,10]],"date-time":"2019-12-10T06:02:35Z","timestamp":1575957755000},"page":"762-780","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["On Minimal Copulas under the Concordance Order"],"prefix":"10.1007","volume":"184","author":[{"given":"Jae Youn","family":"Ahn","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9317-6878","authenticated-orcid":false,"given":"Sebastian","family":"Fuchs","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2019,12,10]]},"reference":[{"key":"1618_CR1","volume-title":"Principles of Copula Theory","author":"F Durante","year":"2016","unstructured":"Durante, F., Sempi, C.: Principles of Copula Theory. 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