An Inscribing Model for Random Polytopes (original) (raw)
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European Journal of Combinatorics, 2007
For convex bodies K with C 2 boundary in R d , we provide results on the volume of random polytopes with vertices chosen along the boundary of K which we call random inscribing polytopes. In particular, we prove results concerning the variance and higher moments of the volume, as well as show that the random inscribing polytopes generated by the Poisson process satisfy central limit theorem.
On the convex hull of uniform random points in a simpled-polytope
Discrete & Computational Geometry, 1991
Denote the expected number of facets and vertices and the expected volume of the convex hull Pn of n random points, selected independently and uniformly from the interior of a simple d-polytope by En(f), E.(v), and E~(V), respectively. In this note we determine the sharp constants of the asymptotic expansion of En(f), E.(v), and En(V), as n tends to infinity. Further, some results concerning the expected shape of P~ are given.
On the Convex Hull of Random Points in a Polytope
Journal of Applied Probability, 1988
The convex hull of n points drawn independently from a uniform distribution on the interior of a d-dimensional polytope is investigated. It is shown that the expected number of vertices is O(log d–1 n) for any polytope, the expected number of vertices is Ω(log d–1 n) for any simple polytope, and the expected number of facets is O(log d–1 n) for any simple polytope. An algorithm is presented for constructing the convex hull of such sets of points in linear average time.
On the Graph-Density of Random 0/1-Polytopes
Lecture Notes in Computer Science, 2003
Let X d,n be an n-element subset of {0, 1} d chosen uniformly at random, and denote by P d,n := conv X d,n its convex hull. Let ∆ d,n be the density of the graph of P d,n (i.e., the number of one-dimensional faces of P d,n divided by n 2). Our main result is that, for any function n(d), the expected value of ∆ d,n(d) converges (with d → ∞) to one if, for some arbitrary ε > 0, n(d) ≤ (√ 2 − ε) d holds for all large d, while it converges to zero if n(d) ≥ (√ 2 + ε) d holds for all large d.
On the isotropic constant of random polytopes
Advances in Geometry, 2000
Let K be an isotropic 1-unconditional convex body in R n . For every N > n consider N independent random points x1, . . . , xN uniformly distributed in K. We prove that, with probability greater than 1 − C 1 exp(−cn) if N ≥ c 1 n and greater than 1−C 1 exp(−cn/ log n) if n < N < c 1 n, the random polytopes KN := conv ± x1, . . . , ±xN and SN := conv{x1, . . . , xN } have isotropic constant bounded by an absolute constant C > 0.
Limit Theorems for Functionals of Random Convex Hulls in a Unit Disk
Mathematics and Statistics, 2023
In this article, we study the functionals of the convex hull generated by independent observations over two-dimensional random points. When the random points are given in the polar coordinate system, their components are independent of each other, the angular coordinate is distributed uniformly, and the tail of the distribution of the radial coordinate is a regularly varying function near the circle of the unit disk – support. Here, with the pproximation of the binomial point process by an inhomogeneous Poisson one, it is possible to study the asymptotic properties of the main functionals of the convex hull. Using the independence property of the increment of Poisson processes, we find an asymptotic expression for the mean values and variances for the main functionals of the convex hull. Uniform boundedness of exponential moments is proved for the same functionals, in the case when the convex hull is generated from an inhomogeneous Poisson point process inside the disk. The indicated independence property of the increment of the Poisson process allows us to express the area of the convex hull as a sum of independent identically distributed random variables, with which we prove the central limiting theorem for the number of vertices and the area of the convex hull. From the results obtained, we can conclude that if the tail of the distribution near the boundary is heavier, then there are many elements of the sample near the support boundary, and this means that there are many vertices of the convex hull, but the area bounded by the perimeter of the convex hull and the circle, as well as the difference between the perimeter of the convex hull and the circle, becomes negligible.
Convex hulls of uniform samples from a convex polygon
Advances in Applied Probability, 2012
In Groeneboom (1988) a central limit theorem for the number of vertices Nn of the convex hull of a uniform sample from the interior of convex polygon was derived. To be more precise, it was shown that {Nn − 2 3 r log n}/{ 10 27 r log n} 1/2 converges in law to a standard normal distribution, if r is the number of vertices of the convex polygon from which the sample is taken.
Few points to generate a random polytope
1997
A random polytope, K n , is the convex hull of n points chosen randomly, independently, and uniformly from a convex body K^R d. It is shown here that, with high probability, K n can be obtained by taking the convex hull of m = o(n) points chosen independently and uniformly from a small neighbourhood of the boundary of K.