The paper gives a new interpretation and a possible optimization of the well-known $k$-means algorithm for searching for a locally optimal partition of the set $𝒜=\{a_i\in {ℝ}^n:i=1,\cdots ,m\}$ which consists of $k$ disjoint nonempty subsets ${\pi }_1,\cdots ,{\pi }_k$, $1\le k\le m$. For this purpose, a new divided $k$-means algorithm was constructed as a limit case of the known smoothed $k$-means algorithm. It is shown that the algorithm constructed in this way coincides with the $k$-means algorithm if during the iterative procedure no data points appear in the Voronoi diagram....