[{"oa":1,"publication_identifier":{"eissn":["1572-9168"],"issn":["0046-5755"]},"date_published":"2023-11-17T00:00:00Z","type":"journal_article","status":"public","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","main_file_link":[{"url":"https://doi.org/10.1007/s10711-023-00862-3","open_access":"1"}],"month":"11","article_number":"15","oa_version":"Published Version","publication":"Geometriae Dedicata","language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"We introduce a hierachy of equivalence relations on the set of separated nets of a given Euclidean space, indexed by concave increasing functions ϕ:(0,∞)→(0,∞). Two separated nets are called ϕ-displacement equivalent if, roughly speaking, there is a bijection between them which, for large radii R, displaces points of norm at most R by something of order at most ϕ(R). We show that the spectrum of ϕ-displacement equivalence spans from the established notion of bounded displacement equivalence, which corresponds to bounded ϕ, to the indiscrete equivalence relation, coresponding to ϕ(R)∈Ω(R), in which all separated nets are equivalent. In between the two ends of this spectrum, the notions of ϕ-displacement equivalence are shown to be pairwise distinct with respect to the asymptotic classes of ϕ(R) for R→∞. We further undertake a comparison of our notion of ϕ-displacement equivalence with previously studied relations on separated nets. Particular attention is given to the interaction of the notions of ϕ-displacement equivalence with that of bilipschitz equivalence."}],"doi":"10.1007/s10711-023-00862-3","arxiv":1,"day":"17","isi":1,"external_id":{"arxiv":["2102.13046"],"isi":["001105681500001"]},"date_updated":"2024-01-11T13:06:32Z","year":"2023","citation":{"ieee":"M. Dymond and V. Kaluza, “Divergence of separated nets with respect to displacement equivalence,” <i>Geometriae Dedicata</i>. Springer Nature, 2023.","chicago":"Dymond, Michael, and Vojtech Kaluza. “Divergence of Separated Nets with Respect to Displacement Equivalence.” <i>Geometriae Dedicata</i>. Springer Nature, 2023. <a href=\"https://doi.org/10.1007/s10711-023-00862-3\">https://doi.org/10.1007/s10711-023-00862-3</a>.","ama":"Dymond M, Kaluza V. Divergence of separated nets with respect to displacement equivalence. <i>Geometriae Dedicata</i>. 2023. doi:<a href=\"https://doi.org/10.1007/s10711-023-00862-3\">10.1007/s10711-023-00862-3</a>","apa":"Dymond, M., &#38; Kaluza, V. (2023). Divergence of separated nets with respect to displacement equivalence. <i>Geometriae Dedicata</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s10711-023-00862-3\">https://doi.org/10.1007/s10711-023-00862-3</a>","ista":"Dymond M, Kaluza V. 2023. Divergence of separated nets with respect to displacement equivalence. Geometriae Dedicata., 15.","short":"M. Dymond, V. Kaluza, Geometriae Dedicata (2023).","mla":"Dymond, Michael, and Vojtech Kaluza. “Divergence of Separated Nets with Respect to Displacement Equivalence.” <i>Geometriae Dedicata</i>, 15, Springer Nature, 2023, doi:<a href=\"https://doi.org/10.1007/s10711-023-00862-3\">10.1007/s10711-023-00862-3</a>."},"acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria). This work was started while both authors were employed at the University of Innsbruck and enjoyed the full support of Austrian Science Fund (FWF): P 30902-N35. It was continued when the first named author was employed at University of Leipzig and the second named author was employed at Institute of Science and Technology of Austria, where he was supported by an IST Fellowship.","title":"Divergence of separated nets with respect to displacement equivalence","publication_status":"epub_ahead","date_created":"2021-07-14T07:01:27Z","department":[{"_id":"UlWa"}],"article_processing_charge":"Yes (via OA deal)","author":[{"full_name":"Dymond, Michael","last_name":"Dymond","first_name":"Michael"},{"first_name":"Vojtech","last_name":"Kaluza","orcid":"0000-0002-2512-8698","full_name":"Kaluza, Vojtech","id":"21AE5134-9EAC-11EA-BEA2-D7BD3DDC885E"}],"_id":"9651","scopus_import":"1","article_type":"original","publisher":"Springer Nature","quality_controlled":"1"},{"oa_version":"None","month":"10","publication":"Geometriae Dedicata","language":[{"iso":"eng"}],"publication_identifier":{"eissn":["1572-9168"],"issn":["0046-5755"]},"publist_id":"2043","date_published":"1989-10-01T00:00:00Z","type":"journal_article","main_file_link":[{"url":"https://link.springer.com/article/10.1007/BF00181432"}],"status":"public","user_id":"ea97e931-d5af-11eb-85d4-e6957dddbf17","publication_status":"published","article_processing_charge":"No","date_created":"2018-12-11T12:06:49Z","title":"Circles through two points that always enclose many points","intvolume":"        32","_id":"4080","scopus_import":"1","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","last_name":"Edelsbrunner"},{"full_name":"Hasan, Nany","first_name":"Nany","last_name":"Hasan"},{"first_name":"Raimund","last_name":"Seidel","full_name":"Seidel, Raimund"},{"full_name":"Shen, Xiao","last_name":"Shen","first_name":"Xiao"}],"issue":"1","publisher":"Springer","article_type":"original","page":"1 - 12","quality_controlled":"1","doi":"10.1007/BF00181432","day":"01","abstract":[{"lang":"eng","text":"This paper proves that any set of n points in the plane contains two points such that any circle through those two points encloses at least n12−112+O(1)n47  points of the set. The main ingredients used in the proof of this result are edge counting formulas for k-order Voronoi diagrams and a lower bound on the minimum number of semispaces of size at most k."}],"date_updated":"2022-02-14T09:55:28Z","year":"1989","citation":{"ista":"Edelsbrunner H, Hasan N, Seidel R, Shen X. 1989. Circles through two points that always enclose many points. Geometriae Dedicata. 32(1), 1–12.","mla":"Edelsbrunner, Herbert, et al. “Circles through Two Points That Always Enclose Many Points.” <i>Geometriae Dedicata</i>, vol. 32, no. 1, Springer, 1989, pp. 1–12, doi:<a href=\"https://doi.org/10.1007/BF00181432\">10.1007/BF00181432</a>.","short":"H. Edelsbrunner, N. Hasan, R. Seidel, X. Shen, Geometriae Dedicata 32 (1989) 1–12.","ieee":"H. Edelsbrunner, N. Hasan, R. Seidel, and X. Shen, “Circles through two points that always enclose many points,” <i>Geometriae Dedicata</i>, vol. 32, no. 1. Springer, pp. 1–12, 1989.","chicago":"Edelsbrunner, Herbert, Nany Hasan, Raimund Seidel, and Xiao Shen. “Circles through Two Points That Always Enclose Many Points.” <i>Geometriae Dedicata</i>. Springer, 1989. <a href=\"https://doi.org/10.1007/BF00181432\">https://doi.org/10.1007/BF00181432</a>.","apa":"Edelsbrunner, H., Hasan, N., Seidel, R., &#38; Shen, X. (1989). Circles through two points that always enclose many points. <i>Geometriae Dedicata</i>. Springer. <a href=\"https://doi.org/10.1007/BF00181432\">https://doi.org/10.1007/BF00181432</a>","ama":"Edelsbrunner H, Hasan N, Seidel R, Shen X. Circles through two points that always enclose many points. <i>Geometriae Dedicata</i>. 1989;32(1):1-12. doi:<a href=\"https://doi.org/10.1007/BF00181432\">10.1007/BF00181432</a>"},"volume":32,"acknowledgement":"Work on this paper by the first author has been supported by Amoco Fnd. Fac. Dev. Comput. Sci. 1-6-44862 and by the National Science Foundation under Grant CCR-8714565, by the second author has been partially supported by the Digital Equipment Corporation, by the fourth author has been partially supported by the Office of Naval Research under Grant N00014-86K-0416.","extern":"1"}]