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On sets of points with few ordinary hyperplanes
dc.contributor | Ball, Simeon Michael |
dc.contributor.author | Jiménez Izquierdo, Enrique |
dc.contributor.other | Universitat Politècnica de Catalunya. Departament de Matemàtiques |
dc.date.accessioned | 2018-11-14T09:12:12Z |
dc.date.available | 2018-11-14T09:12:12Z |
dc.date.issued | 2018-07 |
dc.identifier.uri | http://hdl.handle.net/2117/124196 |
dc.description.abstract | Let $S$ be a set of $n$ points in the projective $d$-dimensional real space $\mathbb{RP}^d$ such that not all points of $S$ are contained in a single hyperplane and such that any subset of $d$ points of $S$ span a hyperplane. Let an ordinary hyperplane of $S$ be an hyperplane of $\mathbb{RP}^d$ containing exactly $d$ points of $S$. In this paper we study the minimum number of ordinary hyperplanes spanned by any set $S$ of $n$ points in $4$ dimensions, following the work of Ben Green and Terence Tao in the planar version of the problem, as well as the work of Simeon Ball in the $3$ dimensional case. We classify the sets of points in $4$ dimensions that span few ordinary hyperplanes, showing that if $S$ is a set spanning less than $Kn^3$ ordinary hyperplanes, for some $K = o(n^{\frac{1}{6}})$, then all but $O(K)$ points of $S$ must be contained in the intersection of $5$ linearly independent quadrics. |
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 | Discrete Geometry |
dc.subject.other | Incidence and Arrangement Problems |
dc.subject.other | Sylvester-Gallai-Type Problems |
dc.subject.other | Computational Geometry |
dc.subject.other | Combinatorial Geometry |
dc.title | On sets of points with few ordinary hyperplanes |
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-1683 |
dc.rights.access | Open Access |
dc.date.updated | 2018-07-18T06:33:24Z |
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) |