I would like to learn some convex and discrete geometry (number 52 in MSC2010). I thought that it would be interesting to approach it from some other parts of mathematics - either by learning applications of convex and discrete geometry to these parts or (preferably) the other way round.

Question:What are some interesting, non-trivial, preferably "important" results or problems lying in the intersection of convex and discrete geometry with other parts of mathematics?

I am most interested in "other parts" being topology (including algebraic and geometric topology), order theory, logic (including model theory). However, other disciplines are also welcome.

I did some MathSciNet searches, but I am not sure which papers could be interesting.