I'm looking for a text exploring/explaining an as-axiomatic/algebraic/categorical-as-possible approach to convex geometry.
Everytime I open a book about convex polytopes (e.g. Ziegler, Grünbaum) I feel like I'm missing some neat axiomatic picture.
For instance, there is some kind of duality:
point <-> hyperplane
convex hull <-> intersection
V-polytope <-> H-polytope
which essentially has to do with polars I guess. But once you want to actually write clean statements out of that, you have all sorts of papercuts related to e.g. full-dimensionality, being centered around zero, etc.
I guess to make such an approach work, maybe it's better to work "projectively" (i.e. $\mathbb{R}^n$ embedded in $\mathbb{R}^{n+1}$ at height one).
The point is: Can one get a nice clean perspective on convex geometry that way?
Edit. After thinking some more about it, it may be that a better approach than "convex subsets" is actually "convex functions". Then maybe we can look at convex functions from some space, and convex maps from $\mathbb{R}$ to the space, and hopefully those dual notions would match the duality mentionned above?
