# Invariance of K-medoids clustering under distance measure

Suppose we have n data points $$X_1,X_2,...X_n$$ where $$X_i \in \mathbb{R^p}$$ and we are performing k-medoids clustering to this dataset. Will the iterative (PAM) algorithm with identical initialization give the same cluster results between the choice of distance measure as sum of absolute distances or $$L_1$$ norm vs sum of squared distances or squared $$L_2$$ norm?

The assignment step will be identical between the two measures as points relatively close to chosen cluster medoid in absolute distance will still be close in squared distance, but I am not very sure about the cluster medoid re-assignment step. I am trying to come up with a counterexample to demonstrate different results but unable to find anything. Would appreciate any help on this question.

If the question was for k-means instead of k-medoid, I believe the answer would be yes, the cluster results will be identical as distance measure is only used during the assignment step and any monotone transformation would preserve the ordering. Let me know if I am missing something here.