# What happens when $k=1$ in k-means? What's the optimized value of distance for $k=1$?

What is the optimized value of distance $$V(x,c)$$ when $$k=1$$ (number of clusters) in k-means? What is the centroid such that it is optimal?

$$V(x,c) = \sum_j \sum_{x_i \rightarrow c_j} D(x_i,c_j)^2$$

which is the sum of square of the distances of each point to the center of the cluster it is assigned to. This cost function depends on the distance $$D$$ used.

Example: Let $$X=(X_1, X_2, X_3)$$ have $$\bar{X}_1=3$$, $$\bar{X}_2=-13$$, and $$\bar{X}_3=-5$$. The the multivariate mean is $$\bar{X} = (3, -13, -5)$$.
(I think this assumes using $$L_2$$ distance, the usual sense of distance.)