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Large convex holes in random point sets
[article]

2012
*
arXiv
*
pre-print

We show

arXiv:1206.0805v1
fatcat:te6rogq4bzdpzo73g6ouuvalum
*that**the*expected number of vertices of*the*largest*convex*hole of*a*set of*n**random**points*chosen independently and uniformly over R is Θ(*n*/(*n*)), regardless of*the*shape of R. ...*A**convex*hole (or empty*convex*polygon) of*a**point*set P*in**the*plane is*a**convex*polygon with vertices*in*P, containing no*points*of P*in*its interior. Let R be*a*bounded*convex*region*in**the*plane. ... Valtr [23] proved*that**the**probability**that*r*points*chosen at*random*285*in**a**triangle**are**in**convex**position*is 2 r (3r − 3)!/ ((r − 1)!) 3 (2r)! . Using 286*the*bounds (s/e) s < s! ...##
###
Empty non-convex and convex four-gons in random point sets

2015
*
Studia scientiarum mathematicarum Hungarica (Print)
*

Let S be

doi:10.1556/sscmath.52.2015.1.1301
fatcat:ygdtkbmsqjfvloa2624tohpz2e
*a*set of*n**points*distributed uniformly and independently*in**a**convex*, bounded set*in**the*plane.*A*four-gon is called empty if it contains no*points*of S*in*its interior. ... We show*that**the*expected number of empty non-*convex*four-gons with vertices from S is 12n 2 log*n*+ o(*n*2 log*n*) and*the*expected number of empty*convex*four-gons with vertices from S is Θ(*n*2 ). keywords ... Acknowledgements We thank Günter Rote for*pointing*out*that*our proof for E [*N*4 ] = Θ(*n*2 log*n*) of*the*conference version of this work actually gives*the*exact asymptotic growth of*N*4 , which is stated ...##
###
Large convex holes in random point sets

2013
*
Computational geometry
*

We show

doi:10.1016/j.comgeo.2012.11.004
fatcat:3r5npjdghveblmopmfu2sazteu
*that**the*expected number of vertices of*the*largest*convex*hole of*a*set of*n**random**points*chosen independently and uniformly over R is Θ(log*n*/(log log*n*)), regardless of*the*shape of R. ...*A**convex*hole (or empty*convex*polygon) of*a**point*set P*in**the*plane is*a**convex*polygon with vertices*in*P , containing no*points*of P*in*its interior. ...*The**probability**that*r*points*chosen at*random*from*a**triangle*269*are**in**convex**position*is at most (30/r) 2r , for all sufficiently large r.270 Proof. ...##
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CONVEX CURVES AND A POISSON IMITATION OF LATTICES

2014
*
Mathematika
*

We solve

doi:10.1112/s0025579313000259
fatcat:4z25ni5sr5fhtci7yjwnba5g7e
*a**randomized*version of*the*following open question: is there*a*strictly*convex*, bounded curve γ*in**the*plane such*that**the*number of rational*points*on γ, with denominator*n*, approaches infinity ...*The*main result here is*that*with*probability*1 there exists*a*strictly*convex*, bounded curve γ such*that**the*number of spatial Poisson*points*on γ, with intensity d, approaches infinity with d. ...*Probability**that*on step*n**the**point*Q is chosen from*a**triangle*∆ i equals An sn , where*A**n*is*the*contribution of ∆ to s*n*. ...##
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Probability thatn random points are in convex position

1995
*
Discrete & Computational Geometry
*

We show

doi:10.1007/bf02574070
fatcat:rzej2zscpbgovbiwfjamv7oure
*that**n**random**points*chosen independently and uniformly from*a*parallelogram*are**in**convex**position*with*probability*( ~*n*-1*n*! ...*A*finite set of*points**in**the*plane is called*convex*if its*points**are*vertices of*a**convex*polygon.*In*this paper we show*the*following result: ... Pach for*pointing*out some related results. ...##
###
Page 3956 of Mathematical Reviews Vol. , Issue 89G
[page]

1989
*
Mathematical Reviews
*

Asymptotic results

*are*derived for Prob(d,*n*),*the**probability**that**the**convex*hull of*n*independent uniform*random**points*within*a*d-dimensional ball has exactly*n*vertices, and for V(d,*n*),*the*proportion ... Results*are*also derived for Prob, (d,*n*),*the*corresponding prob- ability*that**the*m*random**points*form*a*k-*convex*set, for which*the**convex*hull of any subset of at most k*points*is*a*face of*the**convex*...##
###
On the number of vertices of random convex polyhedra

1972
*
Periodica Mathematica Hungarica
*

Thus our problem can be formulated so

doi:10.1007/bf02018666
fatcat:cqetau6ecbglbhrpwel3bnq43q
*that*we choose three independently and identically distributed*random**points**in**a*regular*triangle*, what is*the**probability**that**the**random**triangle**the*vertices of ...*The**probability*distribution of*a**random**point*P*in*this*triangle*uniquely determines*the**probability*distribution of*the*projected*point*Q on*the*boundary. ...##
###
Page 7193 of Mathematical Reviews Vol. , Issue 98K
[page]

1998
*
Mathematical Reviews
*

“Subsequently, we consider

*the**probability*pj, (XK )*that**the*con- vex hull of j independent*random**points**in*K is disjoint from*the**convex*hull of*A*further independent*random**points**in*K. ...*A*theorem of Herglotz states*in**the*particular case of*n*= 3*random**points*and*a**convex*quadrilateral K4 with vertices V;, V2, V3, V4*that*A3(Kq) is Ssi3(1 $13 )S24( 1 — S24), where s;; and s34 denote ...##
###
Area and Perimeter of the Convex Hull of Stochastic Points
[article]

2015
*
arXiv
*
pre-print

Given

arXiv:1412.5153v3
fatcat:rwvlmxmqunaizawtoh44spdmay
*a*set P of*n**points**in**the*plane, we study*the*computation of*the**probability*distribution function of both*the*area and perimeter of*the**convex*hull of*a**random*subset S of P. ...*The**random*subset S is formed by drawing each*point*p of P independently with*a*given rational*probability*π_p. ... This model of*random**points*differs from*the*model*in*which*n**points**are*chosen independently at*random**in*some Euclidean region, and questions related to*the*final*positions*of*the**points**are*considered ...##
###
Longest convex chains

2009
*
Random structures & algorithms (Print)
*

Assume X_n is

doi:10.1002/rsa.20269
fatcat:4sbi33gifvemvosksigsut63wi
*a**random*sample of*n*uniform, independent*points*from*a**triangle*T.*The*longest*convex*chain, Y, of X_n is defined naturally.*The*length |Y| of Y is*a**random*variable, denoted by L_n. ...*In*this article, we determine*the*order of magnitude of*the*expectation of L_n. We show further*that*L_n is highly concentrated around its mean, and*that**the*longest*convex*chains have*a*limit shape. ...*pointing*out an error*in**the*earlier version of this paper. ...##
###
Online geometric reconstruction

2006
*
Proceedings of the twenty-second annual symposium on Computational geometry - SCG '06
*

−

doi:10.1145/1137856.1137912
dblp:conf/compgeom/ChazelleS06
fatcat:5jtxpvfgo5axvax2hehonhvzdu
*n*−c , for an arbitrarily large constant c > 0, where*n*is*the*size of*the*input. 2 Unless indicated otherwise, all our algorithms*are**randomized*. ... We provide upper and lower bounds on*the*complexity of online reconstruction for*convexity**in*2D and 3D. 1 Throughout this article, "with high*probability*" is shorthand for: with*probability*at least 1 ... Therefore, with*probability*at least 1 − 1/*n*4 , more than*a*2/3-fraction of*the*values appended to*the*list L*are*δ-good edges. Note*that*all δ-good edges*are**in**convex**position*. ...##
###
Online geometric reconstruction

2011
*
Journal of the ACM
*

−

doi:10.1145/1989727.1989728
fatcat:k3kvivwdzzan5gghoznq2cfhui
*n*−c , for an arbitrarily large constant c > 0, where*n*is*the*size of*the*input. 2 Unless indicated otherwise, all our algorithms*are**randomized*. ... We provide upper and lower bounds on*the*complexity of online reconstruction for*convexity**in*2D and 3D. 1 Throughout this article, "with high*probability*" is shorthand for: with*probability*at least 1 ... Therefore, with*probability*at least 1 − 1/*n*4 , more than*a*2/3-fraction of*the*values appended to*the*list L*are*δ-good edges. Note*that*all δ-good edges*are**in**convex**position*. ...##
###
Edge-Removal and Non-Crossing Perfect Matchings
[article]

2011
*
arXiv
*
pre-print

*In*

*the*second part we establish

*a*lower bound for

*the*case where

*the*2n

*points*

*are*randomly chosen. ... We first address

*the*case where

*the*boundary of

*the*

*convex*hull of

*the*original graph contains at most

*n*+ 1

*points*.

*In*this case we show

*that*

*n*edges can be removed, one more than

*the*general case. ... We shall use

*a*well known theorem from

*the*theory of

*random*

*convex*sets to show

*that*

*the*

*probability*

*that*

*in*one of

*the*relevant lattice

*triangles*there

*are*k

*points*

*in*

*convex*

*position*is very small. ...

##
###
Testing convexity of figures under the uniform distribution

2018
*
Random structures & algorithms (Print)
*

How many uniform and independent samples from

doi:10.1002/rsa.20797
fatcat:y3rvb2yev5bbfi23jm3wj3ok5q
*a*figure*that*is -far from*convex**are*needed to detect*a*violation of*convexity*with*probability*at least 2/3? ... We show*that*Θ( −4/3 ) uniform samples*are*necessary and sufficient for detecting*a*violation of*convexity**in*an -far figure and, equivalently, for testing*convexity*of figures with 1-sided error. ... Bárány and Füredi [3] analyze*the**probability**that**n**points*chosen from*the*d-dimensional unit ball*are**in**the**convex**position*. ...##
###
Largest inscribed rectangles in convex polygons

2012
*
Journal of Discrete Algorithms
*

If

doi:10.1016/j.jda.2012.01.002
fatcat:xx342ul3wbcdlit4gvtolshuyq
*the*order of*the*vertices of*the*polygon is given, we present*a**randomized*algorithm*that*computes an inscribed rectangle with area at least (1 − ) times*the*optimum with*probability*t*in*time O ( 1 ... We consider approximation algorithms for*the*problem of computing an inscribed rectangle having largest area*in**a**convex*polygon on*n*vertices. ... Acknowledgements We thank*the*anonymous referees for their helpful comments. ...
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