On k-convex point sets
نویسندگان
چکیده
We extend the (recently introduced) notion of k-convexity of a two-dimensional subset of the Euclidean plane to finite point sets. A set of n points is ∗Corresponding author. Email addresses: [email protected] (Oswin Aichholzer), [email protected] (Franz Aurenhammer), [email protected] (Thomas Hackl), [email protected] (Ferran Hurtado), [email protected] (Alexander Pilz), [email protected] (Pedro Ramos), [email protected] (Jorge Urrutia), [email protected] (Pavel Valtr), [email protected] (Birgit Vogtenhuber) Partially supported by the ESF EUROCORES programme EuroGIGA ComPoSe, Austrian Science Fund (FWF): I 648-N18. Supported by the Austrian Science Fund (FWF): P23629-N18 ‘Combinatorial Problems on Geometric Graphs’. Partially supported by ESF EUROCORES programme EuroGIGA, CRP ComPoSe: MICINN grant EUI-EURC-2011-4306, by projects MICINN MTM2009-7242 and MINECO MTM2012-30951, and by project Gen. Cat. DGR 2009SGR1040. Recipient of a DOC-fellowship of the Austrian Academy of Sciences at the Institute for Software Technology, Graz University of Technology. Part of this work was done while A.P. was visiting the Departamento de Matemáticas, Universidad de Alcalá, Madrid, Spain. Partially supported by MEC grant MTM2011-22792 and by the ESF EUROCORES programme EuroGIGA, CRP ComPoSe, under grant EUI-EURC-2011-4306. Partially supported by Consejo Nacional de Ciencia y Tecnoloǵıa, CONACYT (Mexico) grant number CB-2012-01-0178379. Preprint submitted to CGTA February 24, 2014 considered k-convex if there exists a spanning (simple) polygonization such that the intersection of any straight line with its interior consists of at most k disjoint intervals. As the main combinatorial result, we show that every npoint set contains a subset of Ω(log n) points that are in 2-convex position. This bound is asymptotically tight. From an algorithmic point of view, we show that 2-convexity of a finite point set can be decided in polynomial time, whereas the corresponding problem on k-convexity becomes NP-complete for any fixed k ≥ 3.
منابع مشابه
Functionally closed sets and functionally convex sets in real Banach spaces
Let $X$ be a real normed space, then $C(subseteq X)$ is functionally convex (briefly, $F$-convex), if $T(C)subseteq Bbb R $ is convex for all bounded linear transformations $Tin B(X,R)$; and $K(subseteq X)$ is functionally closed (briefly, $F$-closed), if $T(K)subseteq Bbb R $ is closed for all bounded linear transformations $Tin B(X,R)$. We improve the Krein-Milman theorem ...
متن کاملFixed points for total asymptotically nonexpansive mappings in a new version of bead space
The notion of a bead metric space is defined as a nice generalization of the uniformly convex normed space such as $CAT(0)$ space, where the curvature is bounded from above by zero. In fact, the bead spaces themselves can be considered in particular as natural extensions of convex sets in uniformly convex spaces and normed bead spaces are identical with uniformly convex spaces. In this paper, w...
متن کاملA convex combinatorial property of compact sets in the plane and its roots in lattice theory
K. Adaricheva and M. Bolat have recently proved that if $,mathcal U_0$ and $,mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $jin {0,1,2}$ and $kin{0,1}$ such that $,mathcal U_{1-k}$ is included in the convex hull of $,mathcal U_kcup({A_0,A_1, A_2}setminus{A_j})$. One could say disks instead of circles.Here we prove the existence of such a $j$ and $k$ ...
متن کاملk-Sets of Convex Inclusion Chains of Planar Point Sets
Given a set V of n points in the plane, we introduce a new number of k-sets that is an invariant of V : the number of k-sets of a convex inclusion chain of V . A convex inclusion chain of V is an ordering (v1, v2, ..., vn) of the points of V such that no point of the ordering belongs to the convex hull of its predecessors. The k-sets of such a chain are then the distinct k-sets of all the subse...
متن کاملSweep Line Algorithm for Convex Hull Revisited
Convex hull of some given points is the intersection of all convex sets containing them. It is used as primary structure in many other problems in computational geometry and other areas like image processing, model identification, geographical data systems, and triangular computation of a set of points and so on. Computing the convex hull of a set of point is one of the most fundamental and imp...
متن کاملOn an empty triangle with the maximum area in planar point sets
We deal with only finite point sets P in the plane in general position. A point set is convex or in convex position if it determines a convex polygon. A convex subset Q of P is said to be empty if no point of P lies inside the convex hull of Q. An empty convex subset of P with k elements is also called a k-hole of P . Let P be an n planar point set in general position. For a subset Q of P , den...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Comput. Geom.
دوره 47 شماره
صفحات -
تاریخ انتشار 2014