In the support vector machine model, we want to find a plane, $p=w^Tx+b$ that separates two sets of data (labeled positive and negative) with as large a margin as possible. Referencing the notes here, we see on page 7 the objective function:
$$\min_{w,b} \frac 1 2 w^Tw$$ s.t. $$y^{(i)}(w^Tx^{(i)}+b)\geq 1$$
Now, we can use geometric arguments to see that there must be atleast two points on either side of the separating plane for which the inequalities above are "tight" (i.e. they become equalities). I'm wondering - what if I was just given the optimization problem and had no idea about the geometric interpretation. Would it still be possible to somehow extract this fact? Feel free to use the dual form of the problem on page 12.
Here is how the geometric argument works: Imagine the separating plane has a point that is closest to it on the positive side at distance $d$. The point nearest to it on the negative side is at distance $d+\epsilon$. We can improve the objective function, which is the margin by moving the plane (without changing $w$, only $b$) so that it is $\frac{\epsilon}{2}$ close to the closest negative point.