Now that I have external evidence that someone is still hoping for new posts, I thought I better write one.
Here’s a result that not only would have I not have guessed, but I would have assumed the opposite is obviously true. There are convex polytopes that cannot be presented in
The survey paper Non-rational configurations, polytopes, and surfaces, by GÃ¼nter M. Ziegler, gives an explicit construction in 13 dimensions. The paper provides a decent overview of many other, related, topics.