If I guessed really well, hyperspherical will mean that the clusters generated by
k-means are all spheres and by adding more elements/observations to the cluster the spherical shape of
k-means will be expanding in a way that it can't be reshaped with anything but a sphere.
Then the paper is wrong about that,
even that we use
k-means with bunch of data that can be in millions, we are still able to create a cube or any other shape in a three-dimensional space or higher with this data.
Let's say that I added a filter for the pre-processing step that will make the
observations be filled up into two clusters withing 3D space accordingly to shape a cube for the first cluster and a pyramid for the second one.
you can keep adding observations till you have the previous shapes embodied perfectly.
yes, this might seem extremely hard. However, it is still possible. Therefore, we can't say that
k-means will only generates spheres' shapes.
What I want to point out here is that you can force-shape the clusters if you like. even that it is just useless to do so. but as any other data-mining algorithm,
k-means needs a pre-processing before using it. Thus, we can play around with our
observations to get any shape for our clusters. But once again, THIS IS NOT USEFUL IN ANY CASE.