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Discrete Math."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all ultrametrics. The ${CAT}(0)$ metric of Billera--Holmes--Vogtman arises from the theory of orthant spaces. While its geodesics can be computed by the Owen--Provan algorithm, geodesic triangles are complicated. We show that the dimension of such a triangle can be arbitrarily high. Tropical convexity and the tropical metric exhibit properties that are desirable for geometric statistics, such as geodesics of small depth.<\/jats:p>","DOI":"10.1137\/16m1079841","type":"journal-article","created":{"date-parts":[[2017,9,6]],"date-time":"2017-09-06T13:54:13Z","timestamp":1504706053000},"page":"2015-2038","source":"Crossref","is-referenced-by-count":44,"title":["Convexity in Tree Spaces"],"prefix":"10.1137","volume":"31","author":[{"given":"Bo","family":"Lin","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6642-1479","authenticated-orcid":true,"given":"Bernd","family":"Sturmfels","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaoxian","family":"Tang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ruriko","family":"Yoshida","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,9,6]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2011.06.009"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1186\/1748-7188-8-7"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1007\/s00026-004-0227-1"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/120898115"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2005.06.004"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/140953393"},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"M. 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Venkatasubramanian,\n                      Computing hulls, centerpoints and VC dimension in positive definite spaces\n                      , presented at Algorithms and Data Structures Symposium, New York, NY, 2011; original version athttps:\/\/arxiv.org\/abs\/0912.1580.","DOI":"10.1007\/978-3-642-22300-6_33"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1016\/j.jtbi.2016.05.001"},{"key":"atypb17","first-page":"43","author":"Gawrilow E.","year":"2000","journal-title":"Basel"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1214\/12-AAP899"},{"key":"atypb19","doi-asserted-by":"crossref","first-page":"187","DOI":"10.51286\/albjm\/1196710624","volume":"1","author":"Joswig M.","year":"2007","journal-title":"Albanian J. Math."},{"key":"atypb20","unstructured":"B. Lin and R. Yoshida,\n                      Tropical Fermat-Weber Points\n                      , preprint,https:\/\/arxiv.org\/abs\/1604.04674, 2016."},{"key":"atypb21","unstructured":"A. Lubiw, D. 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