As the title say's: Imagine a full-dimensional convex polytope. Is there a lower bound (or even exact formula) for the minimum number of facets the polytope has?
The lower bound is realized by a simplex, so the answer would be $n+1$ in dimension $n$.
You should be able to prove it by taking $n+1$ extreme points of the convex polytope. That gives you a simplex. Show that adding points to that, you are adding faces.