{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T23:00:12Z","timestamp":1740178812188,"version":"3.37.3"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,5,3]],"date-time":"2024-05-03T00:00:00Z","timestamp":1714694400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,5,3]],"date-time":"2024-05-03T00:00:00Z","timestamp":1714694400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["788183"],"award-info":[{"award-number":["788183"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["Z 342-N31","I 02979-N35"],"award-info":[{"award-number":["Z 342-N31","I 02979-N35"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100014988","name":"Institute of Science and Technology","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100014988","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Appl. and Comput. Topology"],"published-print":{"date-parts":[[2024,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The depth of a cell in an arrangement of <jats:italic>n<\/jats:italic> (non-vertical) great-spheres in <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb {S}}^d$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>S<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mi>d<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the number of great-spheres that pass above the cell. We prove Euler-type relations, which imply extensions of the classic Dehn\u2013Sommerville relations for convex polytopes to sublevel sets of the depth function, and we use the relations to extend the expressions for the number of faces of neighborly polytopes to the number of cells of levels in neighborly arrangements.\n<\/jats:p>","DOI":"10.1007\/s41468-024-00173-w","type":"journal-article","created":{"date-parts":[[2024,5,3]],"date-time":"2024-05-03T11:01:54Z","timestamp":1714734114000},"page":"557-578","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Depth in arrangements: Dehn\u2013Sommerville\u2013Euler relations with applications"],"prefix":"10.1007","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5372-7890","authenticated-orcid":false,"given":"Ranita","family":"Biswas","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6249-0832","authenticated-orcid":false,"given":"Sebastiano","family":"Cultrera\u00a0di\u00a0Montesano","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9823-6833","authenticated-orcid":false,"given":"Herbert","family":"Edelsbrunner","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4201-5775","authenticated-orcid":false,"given":"Morteza","family":"Saghafian","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,5,3]]},"reference":[{"key":"173_CR1","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1007\/s00454-002-2778-4","volume":"29","author":"A Andrzejak","year":"2003","unstructured":"Andrzejak, A., Welzl, E.: In between $$k$$-sets, $$j$$-facets, and $$i$$-faces: $$(i, j)$$-partitions. Discrete Comput. Geom. 29, 105\u2013131 (2003)","journal-title":"Discrete Comput. Geom."},{"key":"173_CR2","unstructured":"Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., Saghafian, M.: Counting cells of order-$$k$$ Voronoi tessellations in $${\\mathbb{R}}^3$$ with Morse theory. In: Proc. 37th Ann. Sympos. Comput. Geom., pp. 16:1\u201316:15 (2021)"},{"key":"173_CR3","unstructured":"Bj\u00f6rner, A.: Topological methods. In: Graham, R., Gr\u00f6tschel, M., Lov\u00e1sz, L. (eds.) Handbook of Combinatorics, pp. 1819\u20131872. North-Holland, Amsterdam (1995)"},{"key":"173_CR4","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1007\/PL00009354","volume":"19","author":"TK Dey","year":"1998","unstructured":"Dey, T.K.: Improved bounds for planar $$k$$-sets and related problems. Discrete Comput. Geom. 19, 373\u2013382 (1998)","journal-title":"Discrete Comput. Geom."},{"key":"173_CR5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61568-9","volume-title":"Algorithms in Combinatorial Geometry","author":"H Edelsbrunner","year":"1987","unstructured":"Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer, Heidelberg (1987)"},{"key":"173_CR6","volume-title":"Computational Topology. An Introduction","author":"H Edelsbrunner","year":"2010","unstructured":"Edelsbrunner, H., Harer, J.L.: Computational Topology. An Introduction. American Mathematical Society, Providence (2010)"},{"key":"173_CR7","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/B978-0-7204-2262-7.50018-1","volume-title":"A Survey of Combinatorial Theory","author":"P Erd\u0151s","year":"1973","unstructured":"Erd\u0151s, P., Lov\u00e1sz, L., Simmons, A., Straus, E.G.: Dissection graphs of planar point sets. In: Srivastava, J.N., et al. (eds.) A Survey of Combinatorial Theory, pp. 139\u2013149. North-Holland, Amsterdam (1973)"},{"key":"173_CR8","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1007\/BF01896768","volume":"27","author":"G Fejes Toth","year":"1976","unstructured":"Fejes Toth, G.: Multiple packing and covering of the plane with circles. Acta Math. Acad. Sci. Hung. 27, 135\u2013140 (1976)","journal-title":"Acta Math. Acad. Sci. Hung."},{"key":"173_CR9","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1006\/aima.1997.1650","volume":"134","author":"R Forman","year":"1998","unstructured":"Forman, R.: Morse theory for cell complexes. Adv. Math. 134, 90\u2013145 (1998)","journal-title":"Adv. Math."},{"key":"173_CR10","volume-title":"Convex Polytopes","author":"B Gr\u00fcnbaum","year":"1967","unstructured":"Gr\u00fcnbaum, B.: Convex Polytopes. Wiley, London (1967)"},{"key":"173_CR11","first-page":"93","volume":"44","author":"EE Kummer","year":"1852","unstructured":"Kummer, E.E.: \u00dcber die Erg\u00e4nzungss\u00e4tze zu den allgemeinen Reciprocit\u00e4tsgesetzen. J. Reine Angew. Math. 44, 93\u2013146 (1852)","journal-title":"J. Reine Angew. Math."},{"key":"173_CR12","first-page":"478","volume":"31","author":"D-T Lee","year":"1982","unstructured":"Lee, D.-T.: On $$k$$-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput. 31, 478\u2013487 (1982)","journal-title":"IEEE Trans. Comput."},{"key":"173_CR13","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1016\/S0012-365X(98)00360-4","volume":"223","author":"J Linhart","year":"2000","unstructured":"Linhart, J., Yang, Y., Philipp, M.: Arrangements of hemispheres and halfspaces. Discrete Math. 223, 217\u2013226 (2000)","journal-title":"Discrete Math."},{"key":"173_CR14","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-76649-0","volume-title":"Using the Borsuk\u2013Ulam Theorem. Lectures on Topological Methods in Combinatorics and Geometry","author":"J Matou\u0161ek","year":"2008","unstructured":"Matou\u0161ek, J.: Using the Borsuk\u2013Ulam Theorem. Lectures on Topological Methods in Combinatorics and Geometry, 2nd edn. Universitext, Springer, Heidelberg (2008)","edition":"2"},{"key":"173_CR15","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1112\/S0025579300002850","volume":"17","author":"P McMullen","year":"1970","unstructured":"McMullen, P.: The maximum numbers of faces of a convex polytope. Mathematika 17, 179\u2013184 (1970)","journal-title":"Mathematika"},{"key":"173_CR16","doi-asserted-by":"crossref","unstructured":"Mulmuley, K.: Dehn\u2013Sommerville relations, upper bound theorem, and levels in arrangements. In: Proc. 9th Ann. Sympos. Comput. Geom., pp. 240\u2013146 (1993)","DOI":"10.1145\/160985.161141"},{"key":"173_CR17","doi-asserted-by":"crossref","unstructured":"Shamos, M.I., Hoey, D.: Closest-point problems. In: Proc. 16th Ann. IEEE Symp. Found. Comput. Sci., pp. 151\u2013162 (1975)","DOI":"10.1109\/SFCS.1975.8"},{"key":"173_CR18","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1007\/s00454-001-0005-3","volume":"26","author":"M Sharir","year":"2001","unstructured":"Sharir, M., Smorodinsky, S., Tardos, G.: An improved bound for $$k$$-sets in three dimensions. Discrete Comput. Geom. 26, 195\u2013204 (2001)","journal-title":"Discrete Comput. Geom."},{"key":"173_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8431-1","volume-title":"Lectures on Polytopes","author":"GM Ziegler","year":"1995","unstructured":"Ziegler, G.M.: Lectures on Polytopes. Springer, New York (1995)"},{"key":"173_CR20","unstructured":"\u017divaljevi\u0107, R.T.: Topological methods in discrete geometry. In: Goodman, J.E., O\u2019Rourke, J., T\u00f3th, C.D. (eds.) Handbook of Discrete and Computational Geometry, 3rd edn., pp. 551\u2013580. CRC Press, Boca Raton (2017)"},{"key":"173_CR21","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1016\/0097-3165(92)90028-S","volume":"61","author":"RT \u017divaljevi\u0107","year":"1992","unstructured":"\u017divaljevi\u0107, R.T., Vre\u0107ica, S.T.: The colored Tverberg\u2019s problem and complexes of injective functions. J. Combin. Theory Ser. A 61, 309\u2013318 (1992)","journal-title":"J. Combin. Theory Ser. A"}],"container-title":["Journal of Applied and Computational Topology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s41468-024-00173-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s41468-024-00173-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s41468-024-00173-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,20]],"date-time":"2024-09-20T15:09:16Z","timestamp":1726844956000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s41468-024-00173-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,3]]},"references-count":21,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,9]]}},"alternative-id":["173"],"URL":"https:\/\/doi.org\/10.1007\/s41468-024-00173-w","relation":{},"ISSN":["2367-1726","2367-1734"],"issn-type":[{"type":"print","value":"2367-1726"},{"type":"electronic","value":"2367-1734"}],"subject":[],"published":{"date-parts":[[2024,5,3]]},"assertion":[{"value":"26 July 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 March 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 March 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 May 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}