dc.contributor Huemer, Clemens dc.contributor.author Torra Clotet, Ferran dc.contributor.other Universitat Politècnica de Catalunya. Departament de Matemàtiques dc.date.accessioned 2019-07-11T12:12:47Z dc.date.available 2019-07-11T12:12:47Z dc.date.issued 2019-07 dc.identifier.uri http://hdl.handle.net/2117/166052 dc.description.abstract The Erdős-Szekeres theorem is a famous result in Discrete geometry that inspired a lot of research and motivated new problems. The theorem states that for every integer n ≥ 3 there is another integer N_0 such that any set of N ≥ N_0 points in general position in the plane contains the vertex set of a convex n-gon. Related is the question on the number of empty (without interior points) convex k-gons, X_k, in a set of n points, for k=3,4,5,... . A known result states that the alternating sum of the X_k's only depends on the number n of points, but not on the precise positions of the points. A proof was given by Pinchasi, Radoičić, and Sharir in 2006. In this thesis we extend this result to the numbers of convex k-gons with l interior points, X_{k,l}, and provide several formulas involving these numbers. All these formulas only depend on the number n of points of the set. The proofs are based on a continuous motion argument. We further show that with this proof technique at most n-2 linearly independent equations for the X_{k,l}'s can be obtained and we provide n-2 such equations. We also obtain several other formulas, building upon a work by Huemer, Oliveros, Pérez-Lantero, and Vogtenhuber. The obtained formulas could further be useful to solve some open problems related to the Erdős-Szekeres theorem. This thesis also surveys several known results and questions related to this classical problem for point sets in the plane. dc.language.iso eng dc.publisher Universitat Politècnica de Catalunya dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/3.0/es/ dc.subject Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria dc.subject.lcsh Discrete geometry dc.subject.other Erdős-Szekeres theorem dc.subject.other Convex polygons dc.subject.other Discrete geometry dc.title Combinatorial properties of convex polygons in point sets dc.type Master thesis dc.subject.lemac Geometria discreta dc.subject.ams Classificació AMS::52 Convex and discrete geometry::52C Discrete geometry dc.identifier.slug FME-1797 dc.rights.access Open Access dc.date.updated 2019-07-10T05:29:45Z dc.audience.educationlevel Màster dc.audience.mediator Universitat Politècnica de Catalunya. Facultat de Matemàtiques i Estadística dc.audience.degree MÀSTER UNIVERSITARI EN MATEMÀTICA AVANÇADA I ENGINYERIA MATEMÀTICA (Pla 2010)
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