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Given an arbitrary metric $d$, I want to define a concave continuous function in terms of $d$. For example if $d$ is the Euclidean metric then it is convex, so $-d$ is concave.

Ideally there would be some function which transforms any metric into a concave continuous function but I don't think that's very likely. Instead I wonder if it is possible classify the kinds of metrics and give a function for each kind of metric, like in the following manner:

Suppose every metric is either concave or convex, then either $d$ or $-d$ is a concave continuous function.

So is there some classification of metrics in terms of (quasi/strict) convexity and concavity?

Alternatively let me know if you have good reason to believe that it is impossible to construct a concave continuous function from an arbitrary metric

Ainsley Pullen
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You need more information about the space in order to make sense of concavity/convexity. For instance, suppose that your metric space $(X,d)$ is geodesic, i.e. for any pair of points $x, y\in X$ there exists an isometric embedding $c: [0, D=d(x,y)]\to (X,d)$ such that $c(0)=x, c(D)=y$. Then a function $f: X\to {\mathbb R}$ is called convex if its composition with every geodesic, $f\circ c$, is a convex function of one variable.

Accordingly, a geodesic metric space $(X,d)$ is called convex if for any pair of geodesics $c_1(t), c_2(t)$, the function $$ t\mapsto d(c_1(t), c_2(t)) $$ is convex. For instance, Hilbert spaces satisfy this condition. More generally, if $(M,g)$ is a complete simply-connected Riemannian manifold of sectional curvature $\le 0$, then the Riemannian distance function $d=d_g$ on $M$ is convex. This notion was used by Busemann to define nonpositive curvature in the setting of general metric spaces.

This is the best way I see how to interpret your request for "transforming" a metric into a convex function. As for a "classification", I have no idea what this means.

Moishe Kohan
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