Few Points to Generate a Random Polytope
نویسندگان
چکیده
A random polytope, Kn, is the convex hull of n points chosen randomly, independently, and uniformly from a convex body K^R. It is shown here that, with high probability, Kn 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. §
منابع مشابه
On the Mean Number of Facets of a Gaussian Polytope
A Gaussian polytope is the convex hull of normally distributed random points in a Euclidean space. We give an improved error bound for the expected number of facets of a Gaussian polytope when the dimension of the space is fixed and the number of points tends to infinity. The proof applies the theory of the asymptotic distribution of the top order statistic of a collection of independently dist...
متن کاملA sweep-plane algorithm for generating random tuples in simple polytopes
A sweep-plane algorithm of Lawrence for convex polytope computation is adapted to generate random tuples on simple polytopes. In our method an affine hyperplane is swept through the given polytope until a random fraction (sampled from a proper univariate distribution) of the volume of the polytope is covered. Then the intersection of the plane with the polytope is a simple polytope with smaller...
متن کاملRecent Results on Random Polytopes
This is a survey over recent asymptotic results on random polytopes in d-dimensional Euclidean space. Three ways of generating a random polytope are considered: convex hulls of finitely many random points, projections of a fixed highdimensional polytope into a random d-dimensional subspace, intersections of random closed halfspaces. The type of problems for which asymptotic results are describe...
متن کاملOn 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 کاملRandom points in halfspheres
A random spherical polytope Pn in a spherically convex set K ⊂ S as considered here is the spherical convex hull of n independent, uniformly distributed random points in K. The behaviour of Pn for a spherically convex set K contained in an open halfsphere is quite similar to that of a similarly generated random convex polytope in a Euclidean space, but the case when K is a halfsphere is differe...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009