Why is it not always possible to compute the centroid of feature vectors?
A metric space consists of a set $X$ of "points" and a metric $d\colon X \times X \to \mathbb{R}_{\geq 0}$ (giving the "distance" between any two points) which satisfies the following constraints:


Symmetry: $d(x,y) = d(y,x)$ for all $x,y \in X$.
Non-triviality: $d(x,y) = 0$ if and only if $x = y$.
Triangle inequality: $d(x,y) \leq d(x,z) + d(z,y)$ for all $x,y,z \in X$.


The definition of centroid doesn't make sense for a general metric space, since you cannot perform arithmetic on the elements of $X$.