# Axiomatic approach to convex geometry

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?

• This could be a place to start: ncatlab.org/nlab/show/convex+space Commented Oct 9, 2020 at 8:35
• Thanks for the link, I didn't think of looking there! Somewhat disappointingly, it doesn't seem the link there actually get to talking about polytopes, faces, etc… I'll look more carefully into it. Commented Oct 9, 2020 at 9:00