And the answer is positive, and I prove it to you.
Suppose you put one cluster center right between those two points over here and
the other two somewhere in here.
It doesn't even have an error.
In your assignment step, you will find that pretty much everything left of
this line would be allocated to the left cluster center.
And as a result,
this is the point where the total rubber band distance is minimized.
So this cluster is very stable.
These two guys over here,
however, separate between themselves the data on the right.
And they will fight for the same data points and
end up somewhere partitioning the cloud on the right side.
And that is a stable solution because in the assignment step, nothing changes.
This guy will still correspond to all the guys over here, and
these guys will correspond to the guys over here.
That's called a local minimum.
And it really depends on the initialization of the cluster centers.
If you had chosen these three cluster centers as your initial guesses,
you would never move away from it.
Thus, make sure it's really important in clustering to be aware of
the fact it's a local here climbing algorithm.
And it can give you suboptimal solutions that, if you divide them again,
it gives you a better solution.
Obviously, in this case with three cluster centers,
you want them over here, over here and just one on the right side over here.