Towards continuous consistency axiom

Development of new algorithms in the area machine learning, especially clustering, comparative studies such as well testing according to software engineering principles requires availability labeled data sets. While standard benchmarks are made available, a broader range sets is necessary order avoid problem overfitting. In this context, theoretical works on axiomatization clustering algorithms, axioms preserving transformations quite cheap way produce from existing ones. However, frequently cited axiomatic system Kleinberg:2002, we show paper, not applicable for finite dimensional Euclidean spaces, which many like $k$-means, operate. particular, so-called outer-consistency axiom fails upon making small changes datapoint positions and inner-consistency valid only identity transformation general settings. Hence propose an alternative system, Kleinberg's inner consistency replaced by centric outer motion axiom. We demonstrate that satisfiable hierarchical version $k$-means with auto-adjusted $k$, hence it contradictory. Additionally, creates convex clusters only, possible create detecting concave still can be satisfied. The practical application may generation test existent ones algorithm testing. %We gravitational replacement does have deficiency.