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Convex transversals

2014
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Computational geometry
*

On the positive side, we give a polynomial-time algorithm to find a

doi:10.1016/j.comgeo.2012.10.009
fatcat:f4ch64gemvamlkb5mrpnagni7m
*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. ...##
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Convex Transversals
[chapter]

2011
*
Lecture Notes in Computer Science
*

On the positive side, we give a polynomial-time algorithm to find a

doi:10.1007/978-3-642-22300-6_5
fatcat:umuhgqo4ereznaojqn77p5taqe
*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. ...##
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Notes on Convex Transversals
[article]

2012
*
arXiv
*
pre-print

In this paper, we prove the problem of stabbing a set of disjoint bends by a

arXiv:1211.5107v1
fatcat:eldipbkmgbcgrmpkm7zefrvzku
*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*. ...##
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Convex Polygons and Common Transversals

2015
*
The American mathematical monthly
*

It is shown that if two planar

doi:10.4169/amer.math.monthly.122.9.836
fatcat:msmughxqxzepxmqsuqi7pp52tu
*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. ...##
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Convex Partial Transversals of Planar Regions

2018
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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

doi:10.4230/lipics.isaac.2018.52
dblp:conf/isaac/KeikhaKKKLSUVW18
fatcat:ceirse5bszdn3amhdqkn25nhd4
*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*. ...##
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Topological transversals to a family of convex sets

2010
*
Discrete & Computational Geometry
*

We prove that for a family F of ρ+k+1 compact

doi:10.1007/s00454-010-9282-z
fatcat:mypfchhp3rgj5iexvqptldisrm
*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*. ...##
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Line Transversals of Convex Polyhedra in ^3
[article]

2008
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arXiv
*
pre-print

We establish a bound of O(n^2k^1+), for any >0, on the combinatorial complexity of the set of line

arXiv:0807.1221v1
fatcat:jsu4fcl2xngbbauvxwn4kf5mbi
*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. ...##
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An Exact Convex Formulation of Optimal Power Flow in Radial Distribution Networks Including Transverse Components
[article]

2017
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arXiv
*
pre-print

In order to overcome these limitations, we propose a

arXiv:1605.01964v6
fatcat:3ld6lhs2ljdajmys64ednbgulq
*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. ...##
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Line Transversals to Translates of a Convex Body

2010
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Discrete & Computational Geometry
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*transversal*. ... Let K be a

*convex*body in the plane. ... Ambrus for many helpful and inspiring discussions on problems about line

*transversals*. ...

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Transversal numbers of translates of a convex body

2006
*
Discrete Mathematics
*

Let F be a family of translates of a fixed

doi:10.1016/j.disc.2006.05.014
fatcat:noo2ccgovnalrjws3bu5kpz5ra
*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. ...##
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Genus zero transverse foliations for weakly convex Reeb flows on the tight 3-sphere
[article]

2022
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arXiv
*
pre-print

We present conditions so that these index-2 orbits are binding orbits of a genus zero

arXiv:2206.12856v3
fatcat:vcge5ye5yzdwjjf6uytana6n7u
*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. ...##
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On the transversal number and VC-dimension of families of positive homothets of a convex body

2010
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Discrete Mathematics
*

First, we find an upper bound on the

doi:10.1016/j.disc.2009.07.030
fatcat:yikouy2brnhidmgieamrnjen64
*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 . ...##
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On the convexity of transversely isotropic chain network models†

2006
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Philosophical Magazine
*

The newly derived framework captures not only

doi:10.1080/14786430500080296
fatcat:pmgkv3bosfdwfef2vpy4oulczi
*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. ...##
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Line Transversals of Convex Polyhedra in $\mathbb{R}^3$

2010
*
SIAM journal on computing (Print)
*

Let P be a family of k

doi:10.1137/080744694
fatcat:4ei42ypusjd33cq7z6kywkwv3y
*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 . ...##
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A colorful theorem on transversal lines to plane convex sets

2008
*
Combinatorica
*

We prove a colorful version of Hadwiger's

doi:10.1007/s00493-008-2385-y
fatcat:vxjhexj6fbbv3bbbngnrye6vta
*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 ...
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