The center of mass (=centroid) is defined as the center point minimizing the squared of the Euclidean distances (=variance). If one of the source point is an outlier corrupting your dataset, and if that outlier goes to infinity, then your centroid follows it can is clearly not a robust centerpoint. The median on the contrary is defined as the center point minimising the sum of Euclidean distances. It is robust as it breaks only if n/2 outlier points go to infinity. However, there is no closed form solutions.
Doing some survey, I found that since Fermat (who allegedly first ask the question for 3 points), it has been studied and rediscovered in many communities. The current labeling of this point should be Fermat-Fagnano-Weber-Torricelli-Steiner point, and I forgot many names…
History is full of insights!