Asymptotic properties of bivariate k-means clusters

A bounded region in R² with a uniform density function defined over it is partitioned into k sub-regions such that the within cluster sum of squares is minimized. An asymptotic (k+∞) lower bound for the within cluster sum of squares of this optimal k-means partition is obtained. This lower bound is useful in suggesting that the graph-configuration of the optimal k-partition would consist of regular hexagons of equal size when k is large enough. An empirical study illustrating these asymptotic properties of blvariate k-means cluster is also presented.