{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,2]],"date-time":"2023-10-02T10:40:12Z","timestamp":1696243212661},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2023,6,13]],"date-time":"2023-06-13T00:00:00Z","timestamp":1686614400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,6,13]],"date-time":"2023-06-13T00:00:00Z","timestamp":1686614400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Swiss Federal Institute of Technology Zurich"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2023,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We study families of depth measures defined by natural sets of axioms. We show that any such depth measure is a constant factor approximation of Tukey depth. We further investigate the dimensions of depth regions, showing that the <jats:italic>Cascade conjecture<\/jats:italic>, introduced by Kalai for Tverberg depth, holds for all depth measures which satisfy our most restrictive set of axioms, which includes Tukey depth. Along the way, we introduce and study a new depth measure called <jats:italic>enclosing depth<\/jats:italic>, which we believe to be of independent interest, and show its relation to a constant-fraction Radon theorem on certain two-colored point sets.<\/jats:p>","DOI":"10.1007\/s00493-023-00045-4","type":"journal-article","created":{"date-parts":[[2023,6,13]],"date-time":"2023-06-13T12:02:57Z","timestamp":1686657777000},"page":"1007-1029","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Enclosing Depth and Other Depth Measures"],"prefix":"10.1007","volume":"43","author":[{"given":"Patrick","family":"Schnider","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,6,13]]},"reference":[{"key":"45_CR1","doi-asserted-by":"crossref","unstructured":"Aloupis, G.: Geometric measures of data depth. In: Data Depth: Robust Multivariate Analysis, Computational Geometry and Applications, pp. 147\u2013158 (2003)","DOI":"10.1090\/dimacs\/072\/10"},{"issue":"1","key":"45_CR2","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1016\/0304-3975(94)00254-G","volume":"147","author":"E Amaldi","year":"1995","unstructured":"Amaldi, E., Kann, V.: The complexity and approximability of finding maximum feasible subsystems of linear relations. Theoret. Comput. Sci. 147(1), 181\u2013210 (1995)","journal-title":"Theoret. Comput. Sci."},{"issue":"9","key":"45_CR3","doi-asserted-by":"publisher","first-page":"703","DOI":"10.1016\/0167-8655(93)90138-4","volume":"14","author":"D Avis","year":"1993","unstructured":"Avis, D.: The m-core properly contains the m-divisible points in space. Pattern Recognit. Lett. 14(9), 703\u2013705 (1993)","journal-title":"Pattern Recognit. Lett."},{"issue":"1","key":"45_CR4","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1007\/s00454-007-9021-2","volume":"39","author":"I B\u00e1r\u00e1ny","year":"2008","unstructured":"B\u00e1r\u00e1ny, I., Hubard, A., Jer\u00f3nimo, J.: Slicing convex sets and measures by a hyperplane. Discret. Comput. Geom. 39(1), 67\u201375 (2008)","journal-title":"Discret. Comput. Geom."},{"issue":"4","key":"45_CR5","doi-asserted-by":"publisher","first-page":"459","DOI":"10.1090\/bull\/1634","volume":"55","author":"I B\u00e1r\u00e1ny","year":"2018","unstructured":"B\u00e1r\u00e1ny, I., Sober\u00f3n, P.: Tverberg\u2019s theorem is 50 years old: a survey. Bull. Am. Math. Soc. 55(4), 459\u2013492 (2018)","journal-title":"Bull. Am. Math. Soc."},{"issue":"3","key":"45_CR6","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1007\/PL00009350","volume":"19","author":"I B\u00e1r\u00e1ny","year":"1998","unstructured":"B\u00e1r\u00e1ny, I., Valtr, P.: A positive fraction Erd\u00f6s-Szekeres theorem. Discret. Comput. Geom. 19(3), 335\u2013342 (1998)","journal-title":"Discret. Comput. Geom."},{"issue":"1","key":"45_CR7","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1007\/BF02564279","volume":"29","author":"DG Bourgin","year":"1955","unstructured":"Bourgin, D.G.: On some separation and mapping theorems. Commentarii Mathematici Helvetici 29(1), 199\u2013214 (1955)","journal-title":"Commentarii Mathematici Helvetici"},{"issue":"5","key":"45_CR8","doi-asserted-by":"publisher","first-page":"566","DOI":"10.1016\/j.comgeo.2012.03.001","volume":"46","author":"D Chen","year":"2013","unstructured":"Chen, D., Morin, P., Wagner, U.: Absolute approximation of Tukey depth: theory and experiments. Comput. Geom. Geom. Optim. 46(5), 566\u2013573 (2013)","journal-title":"Comput. Geom. Geom. Optim."},{"issue":"1","key":"45_CR9","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1070\/SM1994v079n01ABEH003491","volume":"79","author":"VL Dol\u2019nikov","year":"1994","unstructured":"Dol\u2019nikov, V.L.: Transversals of families of sets in in $${\\mathbb{R} }^n$$ and a connection between the Helly and Borsuk theorems. Russ. Acad. Sci. Sbornik Math. 79(1), 93 (1994)","journal-title":"Russ. Acad. Sci. Sbornik Math."},{"issue":"1","key":"45_CR10","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1007\/s00454-017-9893-8","volume":"58","author":"R Fabila-Monroy","year":"2017","unstructured":"Fabila-Monroy, R., Huemer, C.: Caratheodory\u2019s theorem in depth. Discret. Comput. Geom. 58(1), 51\u201366 (2017)","journal-title":"Discret. Comput. Geom."},{"issue":"6","key":"45_CR11","doi-asserted-by":"publisher","first-page":"2199","DOI":"10.1137\/15M1007355","volume":"45","author":"J Fox","year":"2016","unstructured":"Fox, J., Pach, J., Suk, A.: A polynomial regularity lemma for semialgebraic hypergraphs and its applications in geometry and property testing. SIAM J. Comput. 45(6), 2199\u20132223 (2016)","journal-title":"SIAM J. Comput."},{"key":"45_CR12","unstructured":"Har-Peled, S., Zhou., T.: Improved Approximation Algorithms for Tverberg Partitions. arXiv preprint arXiv:2007.08717 (2020)"},{"issue":"3","key":"45_CR13","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1214\/aoms\/1177728498","volume":"26","author":"JL Hodges","year":"1955","unstructured":"Hodges, J.L.: A bivariate sign test. Ann. Math. Stat. 26(3), 523\u2013527 (1955)","journal-title":"Ann. Math. Stat."},{"issue":"1","key":"45_CR14","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1016\/0304-3975(78)90006-3","volume":"6","author":"DS Johnson","year":"1978","unstructured":"Johnson, D.S., Preparata, F.P.: The densest hemisphere problem. Theoret. Comput. Sci. 6(1), 93\u2013107 (1978)","journal-title":"Theoret. Comput. Sci."},{"key":"45_CR15","doi-asserted-by":"crossref","unstructured":"Kalai, G.: Combinatorics with a geometric flavor. Vis. Math. Part II 742 (2000)","DOI":"10.1007\/978-3-0346-0425-3_7"},{"key":"45_CR16","unstructured":"Kalai, G.: Problems in Geometric and Topological Combinatorics. Lecture at FU Berlin (2011)"},{"key":"45_CR17","unstructured":"Kalai, G.: Problems for Imre B\u00e1r\u00e1ny\u2019s Birthday. Discrete Geometry and Convexity in Honour of Imre B\u00e1r\u00e1ny, p. 59 (2017)"},{"issue":"4","key":"45_CR18","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1007\/BF01445182","volume":"57","author":"P Kirchberger","year":"1903","unstructured":"Kirchberger, P.: \u00dcber Tchebychefsche Ann\u00e4herungsmethoden. Mathematische Annalen 57(4), 509\u2013540 (1903)","journal-title":"Mathematische Annalen"},{"issue":"1","key":"45_CR19","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1214\/aos\/1176347507","volume":"18","author":"RY Liu","year":"1990","unstructured":"Liu, R.Y.: On a notion of data depth based on random simplices. Ann. Stat. 18(1), 405\u2013414 (1990)","journal-title":"Ann. Stat."},{"issue":"3","key":"45_CR20","doi-asserted-by":"publisher","first-page":"783","DOI":"10.1214\/aos\/1018031260","volume":"27","author":"RY Liu","year":"1999","unstructured":"Liu, R.Y., Parelius, J.M., Singh, K.: Multivariate analysis by data depth: descriptive statistics, graphics and inference. Ann. Stat. 27(3), 783\u2013858 (1999)","journal-title":"Ann. Stat."},{"key":"45_CR21","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-0039-7","volume-title":"Lectures on Discrete Geometry, Graduate Texts in Mathematics","author":"J Matou\u0161ek","year":"2002","unstructured":"Matou\u0161ek, J.: Lectures on Discrete Geometry, Graduate Texts in Mathematics, vol. 212. Springer, New York (2002)"},{"key":"45_CR22","volume-title":"Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry","author":"J Matou\u0161ek","year":"2007","unstructured":"Matou\u0161ek, J.: Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry. Springer, New York (2007)"},{"key":"45_CR23","doi-asserted-by":"publisher","DOI":"10.1201\/9781315371016","volume-title":"Representation Theory of Symmetric Groups","author":"P-L M\u00e9liot","year":"2017","unstructured":"M\u00e9liot, P.-L.: Representation Theory of Symmetric Groups. Chapman and Hall\/CRC, Boca Raton (2017)"},{"issue":"2","key":"45_CR24","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1023\/A:1023208625954","volume":"13","author":"K Miller","year":"2003","unstructured":"Miller, K., Ramaswami, S., Rousseeuw, P., Sellares, J.A., Souvaine, D., Streinu, I., Struyf, A.: Efficient computation of location depth contours by methods of computational geometry. Stat. Comput. 13(2), 153\u2013162 (2003)","journal-title":"Stat. Comput."},{"key":"45_CR25","first-page":"17","volume-title":"Depth Statistics","author":"K Mosler","year":"2013","unstructured":"Mosler, K.: Depth Statistics, pp. 17\u201334. Springer, Berlin (2013)"},{"issue":"2","key":"45_CR26","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1016\/S0925-7721(97)00022-9","volume":"10","author":"J Pach","year":"1998","unstructured":"Pach, J.: A Tverberg-type result on multicolored simplices. Comput. Geom. 10(2), 71\u201376 (1998)","journal-title":"Comput. Geom."},{"key":"45_CR27","first-page":"291","volume":"21","author":"R Rado","year":"1947","unstructured":"Rado, R.: A theorem on general measure. J. Lond. Math. Soc. 21, 291\u2013300 (1947)","journal-title":"J. Lond. Math. Soc."},{"issue":"3","key":"45_CR28","doi-asserted-by":"publisher","first-page":"238","DOI":"10.1007\/BF02760885","volume":"34","author":"JR Reay","year":"1979","unstructured":"Reay, J.R.: Several generalizations of Tverberg\u2019s theorem. Israel J. Math. 34(3), 238\u2013244 (1979)","journal-title":"Israel J. Math."},{"key":"45_CR29","unstructured":"Rolnick, D., Sober\u00f3n, P.: Algorithms for Tverberg\u2019s theorem via centerpoint theorems. arXiv preprint arXiv:1601.03083 (2016)"},{"issue":"5","key":"45_CR30","doi-asserted-by":"publisher","first-page":"733","DOI":"10.1006\/eujc.2000.0493","volume":"22","author":"J-P Roudneff","year":"2001","unstructured":"Roudneff, J.-P.: Partitions of Points into Simplices with k-Dimensional Intersection. Part I: The Conic Tverberg\u2019s Theorem. Eur. J. Comb. 22(5), 733\u2013743 (2001)","journal-title":"Eur. J. Comb."},{"issue":"5","key":"45_CR31","doi-asserted-by":"publisher","first-page":"745","DOI":"10.1006\/eujc.2000.0494","volume":"22","author":"J-P Roudneff","year":"2001","unstructured":"Roudneff, J.-P.: Partitions of points into simplices with k-dimensional intersection. Part II: Proof of Reay\u2019s conjecture in dimensions 4 and 5. Eur. J. Comb. 22(5), 745\u2013765 (2001)","journal-title":"Eur. J. Comb."},{"issue":"8","key":"45_CR32","doi-asserted-by":"publisher","first-page":"1919","DOI":"10.1016\/j.ejc.2008.12.015","volume":"30","author":"J-P Roudneff","year":"2009","unstructured":"Roudneff, J.-P.: New cases of Reay\u2019s conjecture on partitions of points into simplices with k-dimensional intersection. Eur. J. Comb. 30(8), 1919\u20131943 (2009)","journal-title":"Eur. J. Comb."},{"issue":"3","key":"45_CR33","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1007\/s00454-009-9225-8","volume":"44","author":"W Steiger","year":"2010","unstructured":"Steiger, W., Zhao, J.: Generalized ham-sandwich cuts. Discret. Comput. Geom. 44(3), 535\u2013545 (2010)","journal-title":"Discret. Comput. Geom."},{"key":"45_CR34","volume-title":"Handbook of Discrete and Computational Geometry","author":"CD Toth","year":"2017","unstructured":"Toth, C.D., O\u2019Rourke, J., Goodman, J.E.: Handbook of Discrete and Computational Geometry. Chapman and Hall\/CRC, Boca Raton (2017)"},{"key":"45_CR35","unstructured":"Tukey, J.W.: Mathematics and the picturing of data. In: Proc. International Congress of Mathematicians, pp. 523\u2013531 (1975)"},{"issue":"1","key":"45_CR36","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1112\/jlms\/s1-41.1.123","volume":"1","author":"H Tverberg","year":"1966","unstructured":"Tverberg, H.: A generalization of Radon\u2019s theorem. J. Lond. Math. Soc. 1(1), 123\u2013128 (1966)","journal-title":"J. Lond. Math. Soc."},{"key":"45_CR37","volume-title":"On k-Sets and Applications","author":"U Wagner","year":"2003","unstructured":"Wagner, U.: On k-Sets and Applications. ETH Z\u00fcrich, Z\u00fcrich (2003)"},{"key":"45_CR38","doi-asserted-by":"crossref","unstructured":"Yang, C.-T.: On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujob\u00f4 and Dyson, I. Ann. Math. 262\u2013282 (1954)","DOI":"10.2307\/1969632"},{"issue":"2","key":"45_CR39","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1112\/blms\/22.2.183","volume":"22","author":"RT Zivaljevi\u0107","year":"1990","unstructured":"Zivaljevi\u0107, R.T., Vre\u0107ica, S.T.: An extension of the ham sandwich theorem. Bull. Lond. Math. Soc. 22(2), 183\u2013186 (1990)","journal-title":"Bull. Lond. Math. Soc."},{"issue":"2","key":"45_CR40","first-page":"483","volume":"28","author":"Y Zuo","year":"2000","unstructured":"Zuo, Y., Serfling, R.: Structural properties and convergence results for contours of sample statistical depth functions. Ann. Stat. 28(2), 483\u2013499 (2000)","journal-title":"Ann. Stat."}],"container-title":["Combinatorica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00493-023-00045-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00493-023-00045-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00493-023-00045-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,1]],"date-time":"2023-10-01T19:01:36Z","timestamp":1696186896000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00493-023-00045-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,13]]},"references-count":40,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2023,10]]}},"alternative-id":["45"],"URL":"https:\/\/doi.org\/10.1007\/s00493-023-00045-4","relation":{},"ISSN":["0209-9683","1439-6912"],"issn-type":[{"value":"0209-9683","type":"print"},{"value":"1439-6912","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,13]]},"assertion":[{"value":"10 August 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 April 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 April 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 June 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}