Random Points on the Boundary of Smooth Convex Bodies
نویسنده
چکیده
The convex hull of n independent random points chosen on the boundary of a convex body K ⊂ Rd according to a given density function is a random polytope. The expectation of its i–th intrinsic volume for i = 1, . . . , d is investigated. In the case that the boundary of K is sufficiently smooth, asymptotic expansions for these expected intrinsic volumes as n → ∞ are derived.
منابع مشابه
Approximation 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 Polytopes and Affine Surface Area
Let K be a convex body in R. A random polytope is the convex hull [x1, ..., xn] of finitely many points chosen at random in K. E(K,n) is the expectation of the volume of a random polytope of n randomly chosen points. I. Bárány showed that we have for convex bodies with C3 boundary and everywhere positive curvature c(d) lim n→∞ vold(K)− E(K,n) ( vold(K) n ) 2 d+1 = ∫ ∂K κ(x) 1 d+1 dμ(x) where κ(...
متن کاملIntrinsic Volumes of Inscribed Random Polytopes in Smooth Convex Bodies
LetK be a d-dimensional convex bodywith a twice continuously differentiable boundary and everywhere positive Gauss–Kronecker curvature. Denote byKn the convex hull of n points chosen randomly and independently fromK according to the uniform distribution. Matching lower andupper bounds are obtained for the orders ofmagnitudeof the variances of the sth intrinsic volumes Vs(Kn) of Kn for s ∈ {1, ....
متن کاملRandom Polytopes in Smooth Convex Bodies
Let K<= R be a convex body and choose points xl,x2 xn randomly, independently, and uniformly from K. Then Kn = conv {x, , . . . , *„} is a random polytope that approximates K (as n -») with high probability. Answering a question of Rolf Schneider we determine, up to first order precision, the expectation of vol K -vol Kn when K is a smooth convex body. Moreover, this result is extended to qu...
متن کاملUniversal points of convex bodies and bisectors in Minkowski spaces
We deal with different properties of a smooth and strictly convex body that depend on the behavior of the planar sections of the body parallel to and close to a given tangent plane. The first topic is boundary points where any given convex domain in the tangent plane can be approximated by a sequence of suitably rescaled planar sections (so-called p-universal points). In the second topic, the g...
متن کامل