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On polygons enclosing point sets II
Hurtado Díaz, Fernando Alfredo; Merino, C.; Oliveros, D.; Sakai, T.; Urrutia, J.; Ventura, Inmaculada
Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II; Universitat Politècnica de Catalunya. DCCG - Grup de recerca en geometria computacional, combinatoria i discreta
Let R and B be disjoint point sets such that $R\cup B$ is in general position. We say that B encloses by R if there is a simple polygon P with vertex set B such that all the elements in R belong to the interior of P. In this paper we prove that if the vertices of the convex hull of $R\cup B$ belong to B, and |R| ≤ |Conv(B)| − 1 then B encloses R. The bound is tight. This improves on results of a previous paper in which it was proved that if |R| ≤ 56|Conv (B)| then B encloses R. To obtain our result we prove the next result which is interesting on its own right: Let P be a convex polygon with n vertices $\emph{p_1}$,...,$\emph{p_n}$ and S a set of m points contained in the interior of P, m ≤ n−1. Then there is a convex decomposition {$P_1$,...,$P_n$} of P such that all points from S lie on the boundaries of $P_1$,...,$P_n$, and each $P_i$ contains a whole edge of P on its boundary.
Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria::Geometria convexa i discreta
Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Combinatòria
Convex geometry
Graph theory
Geometria convexa
Grafs, Teoria de
Attribution-NonCommercial-NoDerivs 3.0 Spain

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