assume I correctly find the optimal centroids $c_i$ in the kmeans clustering problem, which finds $k$ centroids that minimizes: $$ \min \sum_i \sum_{x_j\in C_i } \|x_j - \mu_i\|^2 $$where $\mu_i, \ i = 1,2,3,\ldots,k$ are the centroids.
note, the chosen $k$ don't have to be the correct number of components of the underlying mixture distribution, $p(x) = \sum_{i=1,2,\ldots,p}N( x\mid \mu_i,\Sigma_i)$.
The results can be represented by a voronoi graph, for example, for a single 2-d gausssian distribution, it will look something like this:
I wonder what is the relation between the size of each cell and the underlying distribution. Does Kmeans clustering tend to equalize the 'volume' of distribution included in the voronoi cells? for example, the cell at the center of the true gaussian is much smaller than the outliers.
Thanks