How many pieces do $k$ random hyperplanes split space into?

Question

Taking $k$ random hyperplanes $H_1,\dots,H_k$ in $\mathbb{R}^n,$ each $H$ the span of $(n-1)$ vectors iid uniform on the sphere in $\mathbb{R}^n$, then how many connected components does $(\cup_iH_i)^C$ have?

What is the distribution of the area each component occupies on the unit sphere?

For $n=2$ I guess it splits into $2k$ components, with each area proportional to any marginal of a Dirichlet RV with parameters $K=k$ and each $\alpha_i=1/k.$

Answer 1

The question of "how many" is answered in Theorem 1 of "On the number of regions in an $m$-dimensional space cut by $n$ hyperplanes" here where it says (adapted to variable names in the OP):

$\textbf{Theorem 1 }$ The number of regions $C_n(k)$, cut by a $k$-cluster in $\mathbb{R}^n$ (or cut by $k$ great circles in $S^{k-1}$), is exactly twice $G_{n-1}(k-1)$, that is \begin{equation} C_n(k)=2\left[{k-1 \choose 0} + {k-1 \choose 1}+{k-1 \choose 2}+\cdots + {k-1 \choose n-1}\right], \end{equation} where ${a\choose b}$ is 0 if $b>a.$

They define a $k$-cluster to be a collection of $k$ distinct $(n-1)$-dimensional affine sets that have a point in common. But if an affine set characterized by a matrix $M$ and offset $a$, $A=a+\{Mx|x\in\operatorname{dom} M\}$ contains $0$ then there is some $x_0$ where $a+Mx_0=0$ so that $A=a+\{M(x+x_0)|x\in\operatorname{dom}M\}=\{Mx|x\in\operatorname{dom}M\}$ so then $A$ is just a subspace. In particular, a $k$-cluster about 0 is just a collection of $k$ distinct hyperplanes.

Trivially with probablilty 1 the construction in the OP gives $k$ distinct hyperplanes (the collection of outcomes where one of the hyperplane's spanning sets is a linear combination of another has volume 0) so the OP's hyperplanes are a $k$-cluster.

I've moved the discussion of the distribution of area to here: https://mathoverflow.net/questions/293151/what-is-the-shape-of-a-random-finitely-generated-convex-cone