Say $\mathbb{R}^n$ is divided by $k>n$ randomly chosen hyperplanes. Each connected region away from the hyperplanes is the intersection of $k$ half-spaces, so it is a convex cone. It is known that the amount of cones the space is divided into in this situation is $C(k,n)=2\sum_{\ell=0}^{n-1}{k-1\choose \ell}$.
A cone traces a polytope $P$ on the surface of a sphere at the origin. For convenience consider the shape traced on the sphere with surface area $C(k,n)$, so that $P$ is a spherical polytope in $\mathbb{R}^{n-1}$ with average area $1$. Now for large $n$ and $k$:
- How tightly clustered about $1$ is $P$'s volume?
- What is the smallest radius for a ball that covers all but $\varepsilon$% of $P$'s volume?
- What is the largest radius for a ball that is all but $\varepsilon$% covered by $P$?
- Do these statistics hold for almost all of the ensemble of polytopes, or only for any individual $P$?
