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

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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.

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**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.