It is well known that the unit ball $B$ in a norm space must be absorbent, symmetrical with respect to the origin (that is, $B=-B$), convex and also does not contain a subspace of dimension $1$.
More than that, given a real vector space and a set $B$ of said space verifying the four properties stated above, it is possible to define a function (the Minkowski functional) that is a norm. The corresponding unit ball of such a norm is $\cap_{\lambda>1} \lambda B$. If the vector space is $\mathbb{R}^n$, the unit ball is equal to the closure of $B$ in the usual topology.
With this in mind, I want to know what can be said about balls in metric spaces that are also real vector spaces, when the metric does not necessary come from a norm. I know that metric balls need not to be convex. How about the other properties? Is there a version of the result stated in the second paragraph for metric spaces?
