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{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,12,14]],"date-time":"2023-12-14T00:52:02Z","timestamp":1702515122793},"reference-count":13,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2022,8,6]],"date-time":"2022-08-06T00:00:00Z","timestamp":1659744000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,8,6]],"date-time":"2022-08-06T00:00:00Z","timestamp":1659744000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["426054364"]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Beitr Algebra Geom"],"published-print":{"date-parts":[[2023,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We study families of faces for convex semi-algebraic sets via the normal cycle which is a semi-algebraic set similar to the conormal variety in projective duality theory. We propose a convex algebraic notion of a <jats:italic>patch<\/jats:italic>\u2014a term recently coined by Ciripoi, Kaihnsa, L\u00f6hne, and Sturmfels as a tool for approximating the convex hull of a semi-algebraic set. We discuss geometric consequences, both for the semi-algebraic and convex geometry of the families of faces, as well as variations of our definition and their consequences.<\/jats:p>","DOI":"10.1007\/s13366-022-00657-9","type":"journal-article","created":{"date-parts":[[2022,8,6]],"date-time":"2022-08-06T18:02:48Z","timestamp":1659808968000},"page":"851-875","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Families of faces and the normal cycle of a convex semi-algebraic set"],"prefix":"10.1007","volume":"64","author":[{"ORCID":"http:\/\/orcid.org\/0000-0002-3376-6841","authenticated-orcid":false,"given":"Daniel","family":"Plaumann","sequence":"first","affiliation":[]},{"ORCID":"http:\/\/orcid.org\/0000-0001-9456-6889","authenticated-orcid":false,"given":"Rainer","family":"Sinn","sequence":"additional","affiliation":[]},{"given":"Jannik Lennart","family":"Wesner","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,8,6]]},"reference":[{"key":"657_CR1","volume-title":"A Course in Convexity. Graduate Studies in Mathematics","author":"A Barvinok","year":"2002","unstructured":"Barvinok, A.: A Course in Convexity. Graduate Studies in Mathematics, vol. 54. American Mathematical Society, Providence (2002)"},{"key":"657_CR2","doi-asserted-by":"crossref","unstructured":"Blekherman, G., Parrilo, P.A., Thomas, R.R. (eds.) Semidefinite Optimization and Convex Algebraic Geometry, vol. 13. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2013)","DOI":"10.1137\/1.9781611972290"},{"key":"657_CR3","doi-asserted-by":"crossref","unstructured":"Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, Berlin (1998)","DOI":"10.1007\/978-3-662-03718-8"},{"issue":"2","key":"657_CR4","doi-asserted-by":"publisher","first-page":"637","DOI":"10.33044\/revuma.v60n2a22","volume":"60","author":"D Ciripoi","year":"2019","unstructured":"Ciripoi, D., Kaihnsa, N., L\u00f6hne, A., Sturmfels, B.: Computing convex hulls of trajectories. Rev. Uni\u00f3n Mat. Argent. 60(2), 637\u2013662 (2019)","journal-title":"Rev. Uni\u00f3n Mat. Argent."},{"key":"657_CR5","doi-asserted-by":"publisher","volume-title":"Joins and Intersections","author":"H Flenner","year":"1999","unstructured":"Flenner, H., O\u2019Carroll, L., Vogel, W.: Joins and Intersections. Springer, Berlin (1999)","DOI":"10.1007\/978-3-662-03817-8"},{"key":"657_CR6","doi-asserted-by":"crossref","unstructured":"Harris, J.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 133. Springer, New York (1992)","DOI":"10.1007\/978-1-4757-2189-8_11"},{"k
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