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.