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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?

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    $\begingroup$ I'd love to see a clear-cut question. $\endgroup$ – Włodzimierz Holsztyński Apr 23 '14 at 4:02
  • $\begingroup$ A problem with your definition is that it seems to rely at least intuitively on the notion that every three points completely determine a triangle (try that even in a simple space like the Grassmannian of two-planes in four-space to see that it is a strong hypothesis). If this is what you want to model, Matt F. is right in pointing you to the work of Busemann. However notice the strong hypothesis of additivity on the area (cut the triangle in two and the areas of the pieces have to add up to the area of the original). $\endgroup$ – alvarezpaiva Apr 25 '14 at 18:30
  • $\begingroup$ @alvarezpaiva Does your objection not apply to lines and metric spaces too? Even on a sphere we can find two points which don't uniquely determine a geodesic. This hardly makes the definition of metric spaces problematic! $\endgroup$ – Vidit Nanda Apr 25 '14 at 19:06
  • $\begingroup$ @ViditNanda: a lot of spaces we look at do have minimizing geodesics between two points and so the distance (defined only on terms of pairs of points) is related to the length of some curve. However, that was not my point: I merely wished to know what you were trying to model. You talk of triangles and areas and then try to model this by functions on triples of points you must have some idea of what phenomena you are trying to capture or maybe just warn the reader that this is abstraction for its own sake. $\endgroup$ – alvarezpaiva Apr 27 '14 at 6:00
  • $\begingroup$ @alvarezpaiva fair enough: I seek something that simultaneously models (1) geodesic convex hulls in Riemannian manifolds, and (2) correlation complexes. By (2) I mean a simplicial complex whose vertices are events, and d-simplices (for d > 0) are decorated with the correlation of all vertex-events occurring simultaneously. In both cases, the pairwise information imposes constraints on the triple-wise information, and so on. $\endgroup$ – Vidit Nanda Apr 27 '14 at 12:41
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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.

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