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### Large convex holes in random point sets [article]

József Balogh, Hernán González-Aguilar, Gelasio Salazar
2012 arXiv   pre-print
We show 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  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

Ruy Fabila-Monroy, Clemens Huemer, Dieter Mitsche
2015 Studia scientiarum mathematicarum Hungarica (Print)
Let S be 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

József Balogh, Hernán González-Aguilar, Gelasio Salazar
2013 Computational geometry
We show 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.  ...

### CONVEX CURVES AND A POISSON IMITATION OF LATTICES

Nick Gravin, Fedor Petrov, Sinai Robins, Dmitry Shiryaev
2014 Mathematika
We solve 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 .  ...

### Probability thatn random points are in convex position

P. Valtr
1995 Discrete & Computational Geometry
We show 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

A. Prékopa
1972 Periodica Mathematica Hungarica
Thus our problem can be formulated so 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]

Pablo Pérez-Lantero
2015 arXiv   pre-print
Given 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

Gergely Ambrus, Imre Bárány
2009 Random structures & algorithms (Print)
Assume X_n is 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
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
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]

Aviv Sheyn And Ran J. Tessler
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

Piotr Berman, Meiram Murzabulatov, Sofya Raskhodnikova
2018 Random structures & algorithms (Print)
How many uniform and independent samples from 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  analyze the probability that n points chosen from the d-dimensional unit ball are in the convex position.  ...

### Largest inscribed rectangles in convex polygons

Christian Knauer, Lena Schlipf, Jens M. Schmidt, Hans Raj Tiwary
2012 Journal of Discrete Algorithms
If 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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