The mean width of random polytopes circumscribed around a convex body

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.

💡 Research Summary

The paper investigates a fundamental problem in stochastic geometry: given a fixed d‑dimensional convex body (K\subset\mathbb{R}^d), consider the random polytope (K^{(n)}) obtained as the intersection of (n) independent half‑spaces that all contain (K). Each half‑space is determined by a random supporting hyperplane whose outward normal direction (u\in S^{d-1}) and offset (t>h_K(u)) (where (h_K) is the support function of (K)) are drawn from a common probability distribution. Under mild regularity assumptions on this distribution—essentially that the density near the boundary behaves like a power of the excess distance (t-h_K(u))—the authors derive precise asymptotic formulas for two natural geometric quantities of (K^{(n)}):

1. The expected difference of the mean widths, (\mathbb{E}