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Each choice of <jats:italic>K<\/jats:italic> gives a different function on the set of bounded subsets of <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>; we characterize which functions can arise in this way. Our characterization draws on the theory of <jats:italic>diversities<\/jats:italic>, a recently introduced generalization of metrics from functions on pairs to functions on finite subsets. We additionally investigate functions which arise by restricting the generalized circumradius to a finite subset of\u00a0<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>. We obtain elegant characterizations in the case that <jats:italic>K<\/jats:italic> is a simplex or parallelotope.<\/jats:p>","DOI":"10.1007\/s00454-023-00493-1","type":"journal-article","created":{"date-parts":[[2023,4,13]],"date-time":"2023-04-13T14:05:08Z","timestamp":1681394708000},"page":"1862-1883","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Diversities and the Generalized Circumradius"],"prefix":"10.1007","volume":"70","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1963-5535","authenticated-orcid":false,"given":"David","family":"Bryant","sequence":"first","affiliation":[]},{"given":"Katharina T.","family":"Huber","sequence":"additional","affiliation":[]},{"given":"Vincent","family":"Moulton","sequence":"additional","affiliation":[]},{"given":"Paul F.","family":"Tupper","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,4,13]]},"reference":[{"key":"493_CR1","volume-title":"Theory and Applications of Distance Geometry","author":"LM Blumenthal","year":"1970","unstructured":"Blumenthal, L.M.: Theory and Applications of Distance Geometry. 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