  This paper investigates the validity of Kleinberg's axioms for clustering
functions with respect to the quite popular clustering algorithm called
$k$-means. While Kleinberg's axioms have been discussed heavily in the past, we
concentrate here on the case predominantly relevant for $k$-means algorithm,
that is behavior embedded in Euclidean space. We point at some contradictions
and counter intuitiveness aspects of this axiomatic set within $\mathbb{R}^m$
that were evidently not discussed so far. Our results suggest that apparently
without defining clearly what kind of clusters we expect we will not be able to
construct a valid axiomatic system. In particular we look at the shape and the
gaps between the clusters. Finally we demonstrate that there exist several ways
to reconcile the formulation of the axioms with their intended meaning and that
under this reformulation the axioms stop to be contradictory and the real-world
$k$-means algorithm conforms to this axiomatic system.
