Your question is Centroid Update in K-Means. Take a moment with it on the right.
Talk me through your thinking if you like. When you're confident, submit your answer and I'll grade it like a real screen (7/10 or better passes).
Prove mathematically why the mean is the appropriate centroid update for k-means (not just intuitively)
Asked in the round 3 stage. This was a k-means discussion with a principal applied scientist. Reported follow-ups included: “we know as ML engineers that mean is not robust to outliers... but who has proved why mean is equidistant from all data points”
Show the optimization objective for a fixed cluster, derive its minimizer, state the assumptions required, and distinguish the result from robustness and geometric intuition.