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Convex transversals
2014
Computational geometry
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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]
2011
Lecture Notes in Computer Science
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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]
2012
arXiv
pre-print
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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
2015
The American mathematical monthly
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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
2018
International Symposium on Algorithms and Computation
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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
2010
Discrete & Computational Geometry
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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]
2008
arXiv
pre-print
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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]
2017
arXiv
pre-print
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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
2010
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. ...
doi:10.1007/s00454-010-9293-9
fatcat:hvzqbejz4fd2bisfqiqgkqy6ra
Transversal numbers of translates of a convex body
2006
Discrete Mathematics
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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]
2022
arXiv
pre-print
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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
2010
Discrete Mathematics
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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†
2006
Philosophical Magazine
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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 $\mathbb{R}^3$
2010
SIAM journal on computing (Print)
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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
2008
Combinatorica
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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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