Why is it not always possible to compute the centroid of feature vectors? Hi in the data mining and machine learning course that I'm taking there is a subject on feature spaces and there is this part about feature vector aggregation and metric spaces that I don't really understand. Now our curriculum is basically a huge presentation and the whole information about feature vector aggregation fits on one slide, anyway what I don't understand is this:

For a given sample $D$ the centroid can be computed as   $C_D = \frac{1}{|D|}\cdot\sum_{o\in D} o$ 

and then on the slide it says, quote: 


  In a general metric space (that is, not a vector space),
  where we only have pairwise distances, it might not be
  possible to compute a centroid


I don't really think I understand what "only pairwise distance" means (my guess is it has something to do with the fact that a metric space defines the distances between all pairs of elements in its set) or why the consequence of it is that you can't always compute the centroid.

I read somewhere that metric spaces does not have to define addition or scaling and if that is the case it makes sense to me that you can not calculate the centroid. Is this what it essentially is about or am I completely misunderstanding something?
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 $X$.