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Topology"],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we exploit the concept of extended Pareto grid to study the geometric properties of the matching distance for <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {R}^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>R<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-valued regular functions defined on a closed Riemannian manifold. In particular, we prove that in this case the matching distance is realised either at special values or at values corresponding to vertical, horizontal or slope 1 lines.\n<\/jats:p>","DOI":"10.1007\/s41468-023-00128-7","type":"journal-article","created":{"date-parts":[[2023,6,23]],"date-time":"2023-06-23T02:10:05Z","timestamp":1687486205000},"page":"815-830","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Geometry of the matching distance for 2D filtering functions"],"prefix":"10.1007","volume":"7","author":[{"given":"Marc","family":"Ethier","sequence":"first","affiliation":[]},{"given":"Patrizio","family":"Frosini","sequence":"additional","affiliation":[]},{"given":"Nicola","family":"Quercioli","sequence":"additional","affiliation":[]},{"given":"Francesca","family":"Tombari","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,6,22]]},"reference":[{"key":"128_CR1","unstructured":"Bapat, A., Brooks, R., Hacker, C., Landi, C., Mahler, B.I., Stephenson, E.R.: Computing the matching distance of 2-parameter persistence. arXiv:2210.12868 (2022)"},{"issue":"14","key":"128_CR2","doi-asserted-by":"publisher","first-page":"1735","DOI":"10.1016\/j.patrec.2011.07.014","volume":"32","author":"S Biasotti","year":"2011","unstructured":"Biasotti, S., Cerri, A., Frosini, P., Giorgi, D.: A new algorithm for computing the 2-dimensional matching distance between size functions. 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