In a Support Vector Machine, the margins are defined as $$\vec{w}\cdot\vec{x_i}+b\ge1\text{ for }x_i \text{ in class }+1$$ $$\vec{w}\cdot\vec{x_i}+b\le-1\text{ for }x_i \text{ in class }-1$$
I understand that maximizing $\Vert w \Vert$ will maximize the distance between the margins given the constraints, however I'm having a hard time with the following:
The margins are set in such a way that their y-intercepts (in 2D space) will always be 2 units apart. It's easy to see how, with this constraint, the distance between the margins will depend only on their angle, but why would we limit the vertical separation of the margins? Is there no situation in which the optimal margins would have greater vertical separation?
Edit: Horizontal margin lines come to mind. If the ideal margin lines (again, in 2D space) have a slope of zero, then the distance between them might need to be greater than (or perhaps less than) 2. Am I missing something fundamental?