**Update: problem reformulation**
---------------------------------

Following the advice in comments, I now restate my problem using Voronoi
tessellation.

Given a unit hypercube $H\_n=\{(x\_1,\ldots,x\_n)\in \mathbb{R}^n: 0\leq x\_i\leq
1\}$, generate $K$ random points in $H\_n$ using
uniform Poisson point process with intensity 1. 

Let $V\_k$ be the $k$-th Voronoi cell. Define

$\mathcal{V}=\{k\in\{1,\ldots,K\}|L\cap V\_{k}\neq \emptyset\}$,

where $L$ is a line that intersects $H\_n$

### My question

Assume that $L$ is a random line (i.e. $L$ is uniformly distributed over all lines that intersect $[0,1]^n$), what is the expected value of $|\mathcal{V}|$? i.e. what is the average number of Voronoi cell that a random line intersects?

### Clarification of edits

1. In the last formulation, $K$ points were uniformly distributed on $[0,1]^n$. *Anthony Quas* suggests that using a Poisson point process is preferable.
2. My original definition of a random $L$ is: the end points of $L$ are
uniformly distributed on $H\_n$'s edges. However, in *Remark*'s comment:

> 
> I suggest that $L$ should be uniformly distributed over all lines that intersect $[0,1]^n$ . It is not equivalent to the current distribution because with uniform distribution if you know the first intersection point is close to a corner, then the second intersection point is also more likely to be close to that corner.
> 
> 
>

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*I guess this formulation is clearer than the old one. So if you can understand my question by reading above, please ignore everything I wrote below.*

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Old formulation: a messy one
----------------------------

I'll first describe how I divide $\mathbb{R}^{n}$ into $K$ convex sets, then describe my problem.

First of all, I define a function
$\mathcal{C}(\mathbf{x}): \mathbb{R}^{n}\mapsto \{1,\ldots,K\}$ as follows:

$\mathcal{C}(\mathbf{x})=\underset{k=1,\ldots,K}{\mathrm{sargmax}} \mathbf{w}\_{k}^{T}\mathbf{x}$,

where sargmax denotes the maximizer with the smallest $k$ value. Moreover,
define

$S\_{k}=\{\mathbf{x}\in \mathbb{R}^{n}|\mathcal{C}(\mathbf{x})=k\}$,

where $ k\in\{1,\ldots,K\}$.

The following statements can be easily proved.

-The collection $S\_{1},S\_{2},\ldots,S\_{K}$ forms a partition of $\mathbb{R}^{n}$.

-$\forall k\in\{1,\ldots,K\}$, $S\_{k}$ is a convex set.

For a line $L$ in $\mathbb{R}^{n}$, let

$V=\{k\in\{1,\ldots,K\}|L\cap S\_{k}\neq \emptyset\}$.

A *random* line in $\mathbb{R}^n$ is a line connecting two points sampled from a $n$-dimensional normal distribution.

### Here comes my question:

Assume that $\mathbf{w}\_1,\ldots, \mathbf{w}\_K$ are iid random variables in $\mathbb{R}^n$, each $\mathbf{w}\_i \sim \mathcal{N}\_n(\mu, \Sigma)$.
For a random line $L$ in $\mathbb{R}^{n}$, what is the expected value of $|V|$?

It would be also nice if someone could re-formulate this problem into a less cumbersome one, or perhaps point me to some literature about this problem.