In all introductory texts I'm aware of, convex polytopes are dealt with strictly within $\mathbb R^n$. Being used to quite larger levels of generality elsewhere, I'm wondering if the same can be done here. Specifically, I'm looking for a generalization of convex polytopes in other spaces such that their face lattices still satisfy most of the properties they do in $\mathbb R^n$, like being graded lattices and satisfying the diamond property.


An obvious setting in which convex polytopes work are topological vector spaces of finite dimension. Unfortunately all of these are isomorphic to $\mathbb R^n$, so nothing new.


Interestingly, it seems like most if not all of the basic definitions of the theory can be made in terms only of a betweenness relation. Convex sets/hulls, affine subspaces/hulls, and closed or open half-spaces can be defined without even a single axiom attached to this relation. Of course, very little can actually be proved about these notions, and I don't know what axioms would be needed to get "expected behavior".


I believe hyperbolic space might satisfy my constraints. However I see no obvious way to generalize this example.



Convex polytopes in other spaces
Some time ago I was having the same question and I found these two:


* Robert Morelli *"[A theory of polyhedra](https://doi.org/10.1006/aima.1993.1001)"* (Adv. Math. 97, No. 1, 1-73 (1993)). He works in modules over ordered rings (and vector spaces over ordered fields). This was very helpful personally to understand valuations on polyhedral algebras in a more general context. zbMATH review [here](https://zbmath.org/?q=an%3A0779.52016).
* Walter Prenowitz and James Jantosciak *"[Join geometries. A theory of convex sets and linear geometry](https://doi.org/10.1007/978-1-4613-9438-9)"* (Springer UTM). zbMATH review [here](https://zbmath.org/?q=an%3A0421.52001). I haven't looked at this book in detail, but seems interesting (and perhaps related to your "betweenness relation"). The second paragraph from its introduction says:



> 
> The familiar figures of classical geometry $-$points, segments, lines,
> planes, triangles, circles, and so on$-$ stem from problems in the physical
> world and seem to be conceptually unrelated. However, a natural setting
> for their study is provided by the concept of *convex set,* which is comparatively
> new in the history of geometrical ideas. The familiar figures can then
> appear as convex sets, boundaries of convex sets, or finite unions of
> convex sets. Moreover, two basic types of figure in linear geometry are
> special cases of convex set: *linear space* (point, line, and plane) and
> *halfspace* (ray, halfplane, and halfspace). Therefore we choose convex set
> to be the central type of figure in our treatment of geometry. How can the
> wealth of geometric knowledge be organized around this idea? By definition,
> a set is convex if it contains the segment joining each pair of its
> points; that is, if it is closed under the operation of joining two points to
> form a segment. But this is precisely the basic operation in Euclid. Our
> point of departure is to take the operation of joining two points to form a
> segment as fundamental, and to throw the burden of unifying the material
> on the consistent and relentless exploitation of this operation.
> 
> 
> 


