For separable data that contain both positive and negative examples, and a separating hyperplane , define the positive-side margin to be the distance between and the nearest data point of class . Similarly, define the negative-side margin to be the distance between and the nearest data point of class . Argue that if is the optimal hyperplane, then . That is, the thickness of the cushion on either side of the optimal is equal.
Suppose , then:
The nearest positive points then stay in hyperplane , while the nearest negative points stay in hyperplane . More general:
Now consider the hyperplane: , first it is still a seperating hyperplane, quick check:
Second, we have:
So yes, the nearest positive points are still the old nearest positive points, the nearest negative points are still the old the nearest negative points:
The only difference is that: (due to the fact that ). So cannot be the optimal hyperplane if . The case is similar, I guess.