Areas of Triangles in (Non-Riemannian) Metric spaces?

Question

I'm looking for a reasonable way to coherently axiomatize both length and area in the absence of a Riemannian structure, i.e., starting only with a metric space; but it's not clear how much of this would be reinventing the wheel. So I'll try to give an example of the sort of thing I'm searching for in the hopes that someone has seen it already and can tell me where to look for the correct axioms.

Background

Consider a set $X$ and a function $e: X^3 \to \mathbb{R}$ so that

1. $e(x,y,z) \geq 0$ with equality occurring if at least two of $x,y,z$ coincide in $X$,
2. $e$ is invariant under the obvious action of the symmetric group $S_3$ on $X^3$,
3. $e(x,y,z) \leq e(w,y,z) + e(x,w,y) + e(x,y,w)$ for all $w,x,y,z$ in $X$.

As you might imagine, such structures have been defined before (Gahler called them 2-metric spaces in the sixties). They are supposed to axiomatize "area of triangle given three vertices" in the same way that the usual metric space definition axiomatizes "length of line between two points".

Question

Given an ordinary metric space $(X,d)$, the defect in the usual triangle inequality -- given by the Gromov product $(x,y)_z = d(x,z) + d(y,z) - d(x,y)$ -- morally measures deviation from collinearity of $x, y$ and $z$; hence one might expect (again, morally) that if $(x,y)_z > 0$ then the area of the triangle spanning $(x,y,z)$ is also strictly positive.

For each triple of points $x,y,z$ in $X$, let $\delta_{xyz}$ be the minimum of $(x,y)_z, (x,z)_y$ and $(y,z)_x$ and let $\mu_{xyz}$ be the largest of the three pairwise distances $d(x,y), d(y,z)$ and $d(x,z)$. In Euclidean space for instance, assuming I didn't calculate poorly, if we let $e(x,y,z)$ be the area of the triangle between $x,y$ and $z$ then there are constants $\alpha^\pm \geq 0$ so that the following inequalities hold:

$$ \alpha^-\cdot\delta_{xyz}\cdot\mu_{xyz}^3 \leq e^2(x,y,z) \leq \alpha^+\cdot\delta_{xyz}\cdot\mu^3_{xyz} $$

This inequality sandwiches the (square of) the area between products of minimax type quantities involving lengths, thus quantifying the extent to which the side lengths are modulating the area.

Thus, if one enhances a metric space $(X,d)$ with a ternary function $e$ subject to the axioms 1-3 in the background and $d$-compatibility coming from something like this last inequality, then one has a reasonable candidate for "metric-spaces with area". In principle we could also go beyond and introduce a volume function $f:X^4 \to \mathbb{R}$ similarly compatible with $e$.

Have you come across a similar axiom scheme $(X,d,e,\ldots)$ without underlying Riemannian metrics? If so, what was it used for and where can I find it and its properties?

Answer 1

See Herbert Busemann's 1955 book, Geometry of Geodesics, secs. 48 and 50.

He worked in the context of metric spaces with some additional requirements on the distance $xy$. One of his key definitions is that "$y$ is between $x$ and $z$", or $(xyz)$ iff $d(x,y)+d(y,z)=d(x,z)$.

His axioms on area, translated to your notation, are:

1. $e(x,y,z)$ is symmetric in $x, y, z$ and non-negative.

2. $e(x,y,z) = 0$ iff either $(xyz)$ or $(yzx)$ or $(zxy)$

3. If $(xyz)$ then $e(w,x,y)+e(w,y,z)=e(w,x,z)$.

He used such area functions to characterize homogeneous spaces, and to find necessary and sufficient conditions for affine structure on the space.