{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,10,20]],"date-time":"2022-10-20T17:26:29Z","timestamp":1666286789467},"reference-count":22,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2019,1,1]],"date-time":"2019-01-01T00:00:00Z","timestamp":1546300800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We consider <jats:italic>Hausdorff discretization<\/jats:italic> from a metric space <jats:italic>E<\/jats:italic> to a discrete subspace <jats:italic>D<\/jats:italic>, which associates to a closed subset <jats:italic>F<\/jats:italic> of <jats:italic>E<\/jats:italic> any subset <jats:italic>S<\/jats:italic> of <jats:italic>D<\/jats:italic> minimizing the Hausdorff distance between <jats:italic>F<\/jats:italic> and <jats:italic>S<\/jats:italic>; this minimum distance, called the <jats:italic>Hausdorff radius<\/jats:italic> of <jats:italic>F<\/jats:italic> and written <jats:italic>r<jats:sub>H<\/jats:sub>\n                  <\/jats:italic>(<jats:italic>F<\/jats:italic>), is bounded by the resolution of <jats:italic>D<\/jats:italic>. We call a closed set <jats:italic>F separated<\/jats:italic> if it can be partitioned into two non-empty closed subsets <jats:italic>F<\/jats:italic>\n                  <jats:sub>1<\/jats:sub> and <jats:italic>F<\/jats:italic>\n                  <jats:sub>2<\/jats:sub> whose mutual distances have a strictly positive lower bound. Assuming some minimal topological properties of <jats:italic>E<\/jats:italic> and <jats:italic>D<\/jats:italic> (satisfied in \u211d<jats:italic>\n                     <jats:sup>n<\/jats:sup>\n                  <\/jats:italic> and \u2124<jats:italic>\n                     <jats:sup>n<\/jats:sup>\n                  <\/jats:italic>), we show that given a non-separated closed subset <jats:italic>F<\/jats:italic> of <jats:italic>E<\/jats:italic>, for any <jats:italic>r<\/jats:italic> &gt; <jats:italic>r<jats:sub>H<\/jats:sub>\n                  <\/jats:italic>(<jats:italic>F<\/jats:italic>), every Hausdorff discretization of <jats:italic>F<\/jats:italic> is connected for the graph with edges linking pairs of points of <jats:italic>D<\/jats:italic> at distance at most 2<jats:italic>r<\/jats:italic>. When <jats:italic>F<\/jats:italic> is connected, this holds for <jats:italic>r<\/jats:italic> = <jats:italic>r<jats:sub>H<\/jats:sub>\n                  <\/jats:italic>(<jats:italic>F<\/jats:italic>), and its greatest Hausdorff discretization belongs to the partial connection generated by the traces on <jats:italic>D<\/jats:italic> of the balls of radius <jats:italic>r<jats:sub>H<\/jats:sub>\n                  <\/jats:italic>(<jats:italic>F<\/jats:italic>). However, when the closed set <jats:italic>F<\/jats:italic> is separated, the Hausdorff discretizations are disconnected whenever the resolution of <jats:italic>D<\/jats:italic> is small enough. In the particular case where <jats:italic>E<\/jats:italic> = \u211d<jats:italic>\n                     <jats:sup>n<\/jats:sup>\n                  <\/jats:italic> and <jats:italic>D<\/jats:italic> = \u2124<jats:italic>\n                     <jats:sup>n<\/jats:sup>\n                  <\/jats:italic> with norm-based distances, we generalize our previous results for <jats:italic>n<\/jats:italic> = 2. For a norm invariant under changes of signs of coordinates, the greatest Hausdorff discretization of a connected closed set is axially connected. For the so-called <jats:italic>coordinate-homogeneous<\/jats:italic> norms, which include the <jats:italic>L<jats:sub>p<\/jats:sub>\n                  <\/jats:italic> norms, we give an adjacency graph for which all Hausdorff discretizations of a connected closed set are connected.<\/jats:p>","DOI":"10.1515\/mathm-2019-0001","type":"journal-article","created":{"date-parts":[[2019,11,7]],"date-time":"2019-11-07T17:06:02Z","timestamp":1573146362000},"page":"1-28","source":"Crossref","is-referenced-by-count":1,"title":["Correspondence between Topological and Discrete Connectivities in Hausdorff Discretization"],"prefix":"10.1515","volume":"3","author":[{"given":"Christian","family":"Ronse","sequence":"first","affiliation":[{"name":"ICube, Universit\u00e9 de Strasbourg , CNRS, 300 Bd S\u00e9bastien Brant, CS 10413, 67412 Illkirch Cedex France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Loic","family":"Mazo","sequence":"additional","affiliation":[{"name":"ICube, Universit\u00e9 de Strasbourg , CNRS, 300 Bd S\u00e9bastien Brant, CS 10413, 67412 Illkirch Cedex France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohamed","family":"Tajine","sequence":"additional","affiliation":[{"name":"ICube, Universit\u00e9 de Strasbourg , CNRS, 300 Bd S\u00e9bastien Brant, CS 10413, 67412 Illkirch Cedex France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2019,11,4]]},"reference":[{"key":"2022042707592436015_j_mathm-2019-0001_ref_001_w2aab3b7b1b1b6b1ab1ab1Aa","unstructured":"[1] B. 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