K-means clustering uses the sum of squared errors (SSE)
$E = \sum\limits_{i=1}^k \sum\limits_{p \in C_i} (p-m_i)^2$ (with k clusters, C the set of objects in a cluster, m the center point of a cluster)
after each iteration to check if SSE is decreasing, until reaching the local minimum/optimum. The benefit of k-medoid is "It is more robust, because it minimizes a sum of dissimilarities instead of a sum of squared Euclidean distances". Though understanding that further distance of a cluster increases the SSE, I still don't understand why it is needed for k-means but not for k-medoids.
p.s. 1. The original paper on k-means would probably explain such a matter, but how to find? Couldn't find on google scholar, or where else to search? 2. This is my first post, happy for any feedback. Should I go more into detail on k-means before a question (how other do), thought it wouldn't be necessary because if someone doesn't know what it is, a small introduction won't really help.