I know that the set definition of a polytope $P^+_f$ associated with a polymatroid function f is:
$$ P^+_f = \{ y \in \mathbb{R}^V_+ : y(A) \leq f(A), \forall A \subseteq V\}$$
where $y(A) = \sum_{i \in A } y_i$ denotes the component sum using the indices in $A$.
However, what is the general definition of a polytope in terms of set theory?
I know how polytopes look in general because I understand intuitively how polygons (polytopes in 2D) and polyhedra (polytopes in 3D) look like, but I am unsure how to express that intuition in rigorous and precise set theory. Anyone know how? Plus wikipedia doesn't seem that helpful in that regard :(