Abstract
We provide a streamlined proof and improved estimates for the weak multivariate Gnedenko law of large numbers on concentration of random polytopes within the space of convex bodies (in a fixed or a high dimensional setting), as well as a corresponding strong law of large numbers.
Highlights
Let d ∈ N and let μ be a probability measure on Rd with a log-concave density f = dμ/dx, i.e. − log f is a convex extended real valued function
We provide a streamlined proof and improved estimates for the weak multivariate Gnedenko law of large numbers on concentration of random polytopes within the space of convex bodies, as well as a corresponding strong law of large numbers
Is a random polytope and, as such, is a random element w.p.1 of the space Kd of all convex bodies in Rd
Summary
The notion of the expectation of a random convex body follows the theory of integrals of set valued functions, see for example [1, 3, 9, 20, 23] and the references therein. It was used in [2] for the purpose of a Kolmogorov strong law of large numbers and has appeared as an approximant to floating bodies in bounded domains [6], as well as in other contexts e.g.
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