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dc.contributorBall, Simeon Michael
dc.contributor.authorJiménez Izquierdo, Enrique
dc.contributor.otherUniversitat Politècnica de Catalunya. Departament de Matemàtiques
dc.date.accessioned2018-11-14T09:12:12Z
dc.date.available2018-11-14T09:12:12Z
dc.date.issued2018-07
dc.identifier.urihttp://hdl.handle.net/2117/124196
dc.description.abstractLet $S$ 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.isoeng
dc.publisherUniversitat Politècnica de Catalunya
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/3.0/es/
dc.subjectÀrees temàtiques de la UPC::Matemàtiques i estadística::Geometria
dc.subject.lcshDiscrete geometry
dc.subject.otherDiscrete Geometry
dc.subject.otherIncidence and Arrangement Problems
dc.subject.otherSylvester-Gallai-Type Problems
dc.subject.otherComputational Geometry
dc.subject.otherCombinatorial Geometry
dc.titleOn sets of points with few ordinary hyperplanes
dc.typeMaster thesis
dc.subject.lemacGeometria discreta
dc.subject.amsClassificació AMS::52 Convex and discrete geometry::52C Discrete geometry
dc.identifier.slugFME-1683
dc.rights.accessOpen Access
dc.date.updated2018-07-18T06:33:24Z
dc.audience.educationlevelMàster
dc.audience.mediatorUniversitat Politècnica de Catalunya. Facultat de Matemàtiques i Estadística
dc.audience.degreeMÀSTER UNIVERSITARI EN MATEMÀTICA AVANÇADA I ENGINYERIA MATEMÀTICA (Pla 2010)


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