k-mean clustering minimize L1 distance

Question

k-mean clustering minimize L1 norm

In k-mean clustering, if I want to minimize the L1 distance from any point to cluster center, the error function and derivative is shown above. However, according to text book, "Thus, if we solve for cp , we find that cp = median(x ∈ Cp)" Why median is the solution?

Answer 1

The median is the solution because your error is just the sum of the signs (how many higher, how many lower). The median has half the points higher, half the points lower, so that this error vanishes.