{"date_published":"2018-09-01T00:00:00Z","title":"Long geodesics on convex surfaces","doi":"10.1007/s00283-018-9795-5","oa":1,"day":"01","abstract":[{"lang":"eng","text":"The goal of this article is to introduce the reader to the theory of intrinsic geometry of convex surfaces. We illustrate the power of the tools by proving a theorem on convex surfaces containing an arbitrarily long closed simple geodesic. Let us remind ourselves that a curve in a surface is called geodesic if every sufficiently short arc of the curve is length minimizing; if, in addition, it has no self-intersections, we call it simple geodesic. A tetrahedron with equal opposite edges is called isosceles. The axiomatic method of Alexandrov geometry allows us to work with the metrics of convex surfaces directly, without approximating it first by a smooth or polyhedral metric. Such approximations destroy the closed geodesics on the surface; therefore it is difficult (if at all possible) to apply approximations in the proof of our theorem. On the other hand, a proof in the smooth or polyhedral case usually admits a translation into Alexandrov’s language; such translation makes the result more general. In fact, our proof resembles a translation of the proof given by Protasov. Note that the main theorem implies in particular that a smooth convex surface does not have arbitrarily long simple closed geodesics. However we do not know a proof of this corollary that is essentially simpler than the one presented below."}],"date_updated":"2021-01-12T06:47:59Z","date_created":"2018-12-11T11:44:40Z","page":"26 - 31","citation":{"ista":"Akopyan A, Petrunin A. 2018. Long geodesics on convex surfaces. Mathematical Intelligencer. 40(3), 26–31.","mla":"Akopyan, Arseniy, and Anton Petrunin. “Long Geodesics on Convex Surfaces.” Mathematical Intelligencer, vol. 40, no. 3, Springer, 2018, pp. 26–31, doi:10.1007/s00283-018-9795-5.","ama":"Akopyan A, Petrunin A. Long geodesics on convex surfaces. Mathematical Intelligencer. 2018;40(3):26-31. doi:10.1007/s00283-018-9795-5","ieee":"A. Akopyan and A. Petrunin, “Long geodesics on convex surfaces,” Mathematical Intelligencer, vol. 40, no. 3. Springer, pp. 26–31, 2018.","chicago":"Akopyan, Arseniy, and Anton Petrunin. “Long Geodesics on Convex Surfaces.” Mathematical Intelligencer. Springer, 2018. https://doi.org/10.1007/s00283-018-9795-5.","apa":"Akopyan, A., & Petrunin, A. (2018). Long geodesics on convex surfaces. Mathematical Intelligencer. Springer. https://doi.org/10.1007/s00283-018-9795-5","short":"A. Akopyan, A. Petrunin, Mathematical Intelligencer 40 (2018) 26–31."},"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","type":"journal_article","publist_id":"7948","publisher":"Springer","department":[{"_id":"HeEd"}],"scopus_import":1,"publication":"Mathematical Intelligencer","_id":"106","external_id":{"arxiv":["1702.05172"]},"volume":40,"author":[{"id":"430D2C90-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-2548-617X","last_name":"Akopyan","full_name":"Akopyan, Arseniy","first_name":"Arseniy"},{"last_name":"Petrunin","first_name":"Anton","full_name":"Petrunin, Anton"}],"main_file_link":[{"url":"https://arxiv.org/abs/1702.05172","open_access":"1"}],"language":[{"iso":"eng"}],"issue":"3","oa_version":"Preprint","intvolume":" 40","quality_controlled":"1","year":"2018","publication_status":"published","month":"09","status":"public"}