Mean width of random polytopes in a reasonably smooth convex body
نویسندگان
چکیده
منابع مشابه
The mean width of random polytopes circumscribed around a convex body
Let K be a d-dimensional convex body, and let K be the intersection of n halfspaces containing K whose bounding hyperplanes are independent and identically distributed. Under suitable distributional assumptions, we prove an asymptotic formula for the expectation of the difference of the mean widths of K and K, and another asymptotic formula for the expectation of the number of facets of K. Thes...
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 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 کاملVariance asymptotics for random polytopes in smooth convex bodies
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 fo...
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 کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Multivariate Analysis
سال: 2009
ISSN: 0047-259X
DOI: 10.1016/j.jmva.2009.07.003