I have a theoretical problem, I think it is easier to show it instead of talking about it:
So by visual examination we got 2 clusters, which k-means does not necessarily get, because some points on the edge of the bigger cluster are closer to the center of the smaller cluster than to the center of the bigger one. A possible solution would be to move closer to the second cluster with the centroid of the first cluster. I am not sure whether k-means does this (probably not), but if we have multiple small clusters around the big one, this does not help either. How could we modify the k-means to overcome this? Are there clustering algorithms, which don't have this weakness?