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Convex transversals

Esther M. Arkin, Claudia Dieckmann, Christian Knauer, Joseph S.B. Mitchell, Valentin Polishchuk, Lena Schlipf, Shang Yang
2014 Computational geometry  
On the positive side, we give a polynomial-time algorithm to find a convex transversal of a maximum number of pairwise-disjoint segments (or convex polygons) in 2D if the vertices of the transversal are  ...  We prove that when the sets are segments in the plane, deciding existence of the convex stabber is NP-hard. The problem remains NP-hard if the sets are scaled copies of a convex polygon.  ...  Acknowledgments We thank Joe Blitzstein (Harvard University) for pointers to work in convex regression, and the anonymous reviewer for helpful comments. E. Arkin and J.  ... 
doi:10.1016/j.comgeo.2012.10.009 fatcat:f4ch64gemvamlkb5mrpnagni7m

Convex Transversals [chapter]

Esther M. Arkin, Claudia Dieckmann, Christian Knauer, Joseph S. B. Mitchell, Valentin Polishchuk, Lena Schlipf, Shang Yang
2011 Lecture Notes in Computer Science  
On the positive side, we give a polynomial-time algorithm to find a convex transversal of a maximum number of pairwise-disjoint segments (or convex polygons) in 2D if the vertices of the transversal are  ...  We prove that when the sets are segments in the plane, deciding existence of the convex stabber is NP-hard. The problem remains NP-hard if the sets are scaled copies of a convex polygon.  ...  Acknowledgments We thank Joe Blitzstein (Harvard University) for pointers to work in convex regression, and the anonymous reviewer for helpful comments. E. Arkin and J.  ... 
doi:10.1007/978-3-642-22300-6_5 fatcat:umuhgqo4ereznaojqn77p5taqe

Notes on Convex Transversals [article]

Lena Schlipf
2012 arXiv   pre-print
In this paper, we prove the problem of stabbing a set of disjoint bends by a convex stabber to be NP-hard.  ...  We also consider the optimization version of the convex stabber problem and prove this problem to be APX-hard for sets of line segments.  ...  We call this set stabbable if there exists a convex polygon whose boundary intersects every object. The boundary is then called convex stabber or convex transversal.  ... 
arXiv:1211.5107v1 fatcat:eldipbkmgbcgrmpkm7zefrvzku

Convex Polygons and Common Transversals

Péter Hajnal, László I. Szabó, Vilmos Totik
2015 The American mathematical monthly  
It is shown that if two planar convex n-gons are oppositely oriented, then the segments joining the corresponding vertices have a common transversal.  ...  A different formulation is also given in terms of two cars moving along two convex curves in opposite directions.  ...  Figure 1 : 1 The convex polygons and a common transversal of A i B i counterclockwise direction, returning to the starting point at the same time.  ... 
doi:10.4169/amer.math.monthly.122.9.836 fatcat:msmughxqxzepxmqsuqi7pp52tu

Convex Partial Transversals of Planar Regions

Vahideh Keikha, Mees Van De Kerkhof, Marc Van Kreveld, Irina Kostitsyna, Maarten Löffler, Frank Staals, Jérôme Urhausen, Jordi L. Vermeulen, Lionov Wiratma, Michael Wagner
2018 International Symposium on Algorithms and Computation  
We consider the problem of testing, for a given set of planar regions R and an integer k, whether there exists a convex shape whose boundary intersects at least k regions of R.  ...  , the convex hull of a convex transversal forms a convex body whose boundary intersects every set.  ...  Let Q be the maximum convex transversal.  ... 
doi:10.4230/lipics.isaac.2018.52 dblp:conf/isaac/KeikhaKKKLSUVW18 fatcat:ceirse5bszdn3amhdqkn25nhd4

Topological transversals to a family of convex sets

L. Montejano, R. N. Karasev
2010 Discrete & Computational Geometry  
We prove that for a family F of ρ+k+1 compact convex sets in R^d a topological ρ-transversal of index (m,k) implies an ordinary ρ-transversal.  ...  Let F be a family of compact convex sets in R^d.  ...  Let F be a family of ρ + k + 1 compact convex sets in R d . If F has a topological ρ-transversal of index (m, k), then it has an ordinary ρ-transversal.  ... 
doi:10.1007/s00454-010-9282-z fatcat:mypfchhp3rgj5iexvqptldisrm

Line Transversals of Convex Polyhedra in ^3 [article]

Haim Kaplan, Natan Rubin, Micha Sharir
2008 arXiv   pre-print
We establish a bound of O(n^2k^1+), for any >0, on the combinatorial complexity of the set of line transversals of a collection of k convex polyhedra in ^3 with a total of n facets, and present a randomized  ...  To obtain the above result, we study the set of line transversals which emanate from a fixed line ℓ_0, establish an almost tight bound of O(nk^1+) on the complexity of , and provide a randomized algorithm  ...  Let P be a family of k convex sets in R 3 . A line ℓ is a transversal of P if it intersects every member of P.  ... 
arXiv:0807.1221v1 fatcat:jsu4fcl2xngbbauvxwn4kf5mbi

An Exact Convex Formulation of Optimal Power Flow in Radial Distribution Networks Including Transverse Components [article]

Mostafa Nick, Rachid Cherkaoui, Jean-Yves Le Boudec, Mario Paolone
2017 arXiv   pre-print
In order to overcome these limitations, we propose a convex formulation of the OPF problem applied to radial power grids for which the AC-OPF equations, including the transverse parameters, are considered  ...  The A-OPF is not convex due to Equation (8.c). We can make it convex by replacing (8.c) with (10) .  ...  It can be easily shown that R-OPF is a convex problem.  ... 
arXiv:1605.01964v6 fatcat:3ld6lhs2ljdajmys64ednbgulq

Line Transversals to Translates of a Convex Body

J. Jerónimo-Castro, E. Roldán-Pensado
2010 Discrete & Computational Geometry  
transversal.  ...  Let K be a convex body in the plane.  ...  Ambrus for many helpful and inspiring discussions on problems about line transversals.  ... 
doi:10.1007/s00454-010-9293-9 fatcat:hvzqbejz4fd2bisfqiqgkqy6ra

Transversal numbers of translates of a convex body

Seog-Jin Kim, Kittikorn Nakprasit, Michael J. Pelsmajer, Jozef Skokan
2006 Discrete Mathematics  
Let F be a family of translates of a fixed convex set M in R n . Let (F) and (F) denote the transversal number and the independence number of F, respectively.  ...  Furthermore, if M is centrally symmetric convex body in the plane, then (F) (F) 6 (F) − 3.  ...  Fig. 4 . 4 The case when √ 3 r 2. 4 . 4 Transversal number of homothetic copies of a convex set Lemma 12.  ... 
doi:10.1016/j.disc.2006.05.014 fatcat:noo2ccgovnalrjws3bu5kpz5ra

Genus zero transverse foliations for weakly convex Reeb flows on the tight 3-sphere [article]

Naiara V. de Paulo, Umberto Hryniewicz, Seongchan Kim, Pedro A. S. Salomão
2022 arXiv   pre-print
We present conditions so that these index-2 orbits are binding orbits of a genus zero transverse foliation.  ...  A contact form on the tight 3-sphere (S^3,ξ_0) is called weakly convex if the Conley-Zehnder index of every Reeb orbit is at least 2.  ...  A transverse foliation as in Theorem 1.2 is called a weakly convex foliation. In the real-analytic case, its existence implies infinitely many periodic orbits for the Reeb flow of λ.  ... 
arXiv:2206.12856v3 fatcat:vcge5ye5yzdwjjf6uytana6n7u

On the transversal number and VC-dimension of families of positive homothets of a convex body

Márton Naszódi, Steven Taschuk
2010 Discrete Mathematics  
First, we find an upper bound on the transversal number τ(F) of F in terms of n and the independence number ν(F). This question is motivated by a problem of Grünbaum.  ...  Let F be a family of positive homothets (or translates) of a given convex body K in R^n. We investigate two approaches to measuring the complexity of F.  ...  Skokan, Transversal numbers of translates of a convex body, Discrete Math. 306 (18) (2006) 2166-2173] not a transversal of F , continue in this manner, obtaining eventually a transversal of F .  ... 
doi:10.1016/j.disc.2009.07.030 fatcat:yikouy2brnhidmgieamrnjen64

On the convexity of transversely isotropic chain network models†

E. Kuhl, A. Menzel, K. Garikipati
2006 Philosophical Magazine  
The newly derived framework captures not only transversely isotropic chain network effects but also classical isotropic network effects and classical transverse isotropy as special cases.  ...  Nevertheless, in most practical applications, the convexity condition as such might be too difficult 3242 E. Kuhl et al.  ...  The general notions of convexity and rank-one convexity are reiterated in section 4 before a systematic convexity study of the newly developed class of models is carried out in section 5.  ... 
doi:10.1080/14786430500080296 fatcat:pmgkv3bosfdwfef2vpy4oulczi

Line Transversals of Convex Polyhedra in $\mathbb{R}^3$

Haim Kaplan, Natan Rubin, Micha Sharir
2010 SIAM journal on computing (Print)  
Let P be a family of k convex sets in R 3 . A line is a transversal of P if it intersects every member of P.  ...  Line transversals-a brief background. In this paper we study the combinatorial complexity of the set of line transversals of a collection of convex polyhedra in R 3 .  ...  pass through a fixed point, the transversal space has a maximum of Θ(k d−1 ) components for any collection P of k (not necessarily pairwise disjoint) convex sets in R d .  ... 
doi:10.1137/080744694 fatcat:4ei42ypusjd33cq7z6kywkwv3y

A colorful theorem on transversal lines to plane convex sets

Jorge L. Arocha, Javier Bracho, Luis Montejano
2008 Combinatorica  
We prove a colorful version of Hadwiger's transversal line theorem: if a family of colored and numbered convex sets in the plane has the property that any three differently colored members have a transversal  ...  line that meet the sets consistently with the numbering, then there exists a color such that all the convex sets of that color have a transversal line.  ...  Does Hadwiger's Theorem on transversal lines to plane convex sets admits Mathematics Subject Classification (2000) : 52A35 All authors are partially supported by CONACYT research grant 5040017. a colorful  ... 
doi:10.1007/s00493-008-2385-y fatcat:vxjhexj6fbbv3bbbngnrye6vta
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