Perhaps more or less related to my coding efforts, I have taken a new perspective (pun intended) on the definition of polytope. In short, polytope (like creativity) means many things to many people in many contexts. Here are a couple of possible definitions.
If you project your polytope onto a picture plane, it produces a (simpler?) polytope in one fewer dimension. If you consider all possible projections of a polytope, the polytope might be well defined so long as it's projections are well defined. Since polytopes of zero dimension are well defined, this recursive definition might work. I say might, because the projections of a polytope have some relations not captured by simply collecting them into a set.
As another example of defining polytope, start with the usual graph of facets, and add convexity around each facet. My coding experience has increased my respect for directness. On the other hand, without curves we would not know that lines are straight.