The mean width of random polytopes circumscribed around a convex body

TL;DR

This paper derives asymptotic formulas for the expected difference in mean widths and the expected number of facets of random polytopes circumscribed around a convex body, using volume approximation and polarity techniques.

Contribution

It introduces new asymptotic formulas for mean width and facet count of random polytopes around convex bodies under specific distributional assumptions.

Findings

Asymptotic formula for the expectation of mean width difference

Asymptotic formula for the expected number of facets

Results achieved via weighted volume approximation and polarity

Abstract

Let K be a d-dimensional convex body, and let $K^{(n)}$ 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 $K^{(n)}$ and K, and another asymptotic formula for the expectation of the number of facets of $K^{(n)}$. These results are achieved by establishing an asymptotic result on weighted volume approximation of $K$ and by "dualizing" it using polarity.