TITLE: Term for a metric space for which the triangle inequality is strict?
QUESTION [12 upvotes]: Is there a standard term for a metric space in which $\rho(p,r) < \rho(p,q) + \rho(q,r)$ for any distinct $p$, $q$, $r$? Sort of the opposite of metric convexity.
For instance, a subset of euclidean space with the inherited distance function has this property if and only if no three points are colinear. Another example: $\rho^\alpha$ has this property for any metric $\rho$ and any $\alpha \in (0,1)$.
(Context: I'm working on the second edition of my book Lipschitz Algebras, and I realized that a property I called "concavity" in the first edition is equivalent to this simple condition.)

REPLY [3 votes]: The only thing which I recall in this connection is: Blumenthal in his "Theory and Applications of Distance Geometry" (page 56) calls three points a linear triple if the triple is congruent with a triple in $\mathbb{R}^1$. See also page 242 in Blumenthal-Menger, "Studies in Geometry". (Possibly one can find in these books something more relevant, but I do not recall such things now.) So one can call such spaces without linear triples.