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.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
