A polytope is the convex hull of a set of points. Can see immeadately how it would be useful to know if a set of points in Rd form a polytope, linear programming!
How can we formalize this definition?
If we're only thinking about convex polytopes, we could construct a set based on the intersection of the product spaces of the multiplication of each plane through three points and the plane orthogonal to this plane.
P = \cap_{a,b,c \in S} (ab x ac) x (the normal of that plane through a or b or c)
However, this is pretty useless if we're trying to classify things, not to mention computationally stupid.
Rather, we can think of the vectors inside this polytope as the weighted sum of it's vertexs, satisfying the natural conditions that all the wheights sum to 1 and that they are all positive.
Keep wheighted matroids, bases and greedy algorithms in the back of your mind.
In a way, greedy algorithms are also only useful for "convex" problems, even though the "shape" of their "graph" isn't really always convex. Rather, you just know that gradint descent at any point brings you closer to the global optimum.
Well, I guess for all intents and purposes they might as well be the same then.
HEY!
This is a novel way of thinking about polytopes that incorporates the ugly set theoretic definition we came up with earlier.
We can define the number of "faces" as, and I'm tempted to say something like number of homotopy classes (with edges instead of "holes"), but I think it's better to think in terms of closure under some kind of function, it would be some convex linear function in R(d - 1), kinda like a recursive definition, where the faces are themselves polytopes.
Then it's plausible that there exists some relationship between the number of real faces and the number of "closured" faces, and a discrepency between these two indicates a non-convex polytope.
Now I'm thinking this is kinda Hausdorff-ish, no? If you think of a polytopes "dual", where every face is a vertex and every vertex a face, then we say that iff the faces are Hausdorff, then they're convex?
Need to see how these are related, could be interesting.