Variance asymptotics for random polytopes in smooth convex bodies
نویسندگان
چکیده
Let K ⊂ R be a smooth convex set and let Pλ be a Poisson point process on R of intensity λ. The convex hull of Pλ ∩ K is a random convex polytope Kλ. As λ → ∞, we show that the variance of the number of k-dimensional faces of Kλ, when properly scaled, converges to a scalar multiple of the affine surface area of K. Similar asymptotics hold for the variance of the number of k-dimensional faces for the convex hull of a binomial process in K.
منابع مشابه
On the variance of random polytopes
A random polytope is the convex hull of uniformly distributed random points in a convex body K. A general lower bound on the variance of the volume and f -vector of random polytopes is proved. Also an upper bound in the case when K is a polytope is given. For polytopes, as for smooth convex bodies, the upper and lower bounds are of the same order of magnitude. The results imply a law of large n...
متن کاملVariance Asymptotics and Scaling Limits for Random Polytopes
Let K be a convex set in Rd and let Kλ be the convex hull of a homogeneous Poisson point process Pλ of intensity λ on K. When K is a simple polytope, we establish scaling limits as λ → ∞ for the boundary of Kλ in a vicinity of a vertex of K and we give variance asymptotics for the volume and k-face functional of Kλ, k ∈ {0, 1, ..., d − 1}, resolving an open question posed in [18]. The scaling l...
متن کاملBrownian limits, local limits, extreme value and variance asymptotics for convex hulls in the ball
The paper [40] establishes an asymptotic representation for random convex polytope geometry in the unit ball Bd, d ≥ 2, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of the so-called generalized paraboloid growth process. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via ...
متن کاملApproximation of Smooth Convex Bodies by Random Circumscribed Polytopes
Choose n independent random points on the boundary of a convex body K ⊂Rd . The intersection of the supporting halfspaces at these random points is a random convex polyhedron. The expectations of its volume, its surface area and its mean width are investigated. In the case that the boundary of K is sufficiently smooth, asymptotic expansions as n→∞ are derived even in the case when the curvature...
متن کاملRandom inscribing polytopes
For convex bodies K with C2 boundary in Rd, we explore random polytopes with vertices chosen along the boundary of K. In particular, we determine asymptotic properties of the volume of these random polytopes. We provide results concerning the variance and higher moments of this functional, as well as an analogous central limit theorem.
متن کامل