  In this short note, we study the properties of the weighted Frechet mean as a
convex combination operator on an arbitrary metric space, (Y,d). We show that
this binary operator is commutative, non-associative, idempotent, invariant to
multiplication by a constant weight and possesses an identity element. We also
treat the properties of the weighted cumulative Frechet mean. These tools allow
us to derive several types of median inequalities for abstract metric spaces
that hold for both negative and positive Alexandrov spaces. In particular, we
show through an example that these bounds cannot be improved upon in general
metric spaces. For weighted Frechet means, however, such inequalities can
solely be derived for weights equal or greater than one. This latter limitation
highlights the inherent difficulties associated with working with
abstract-valued random variables.
