Assume contains two data points and . Show that:
(a) No hyperplane can tolerate noise radius greater than .
Assume such hyperplane exists and we call it .
Let is the line connecting two points and , as two points stay on different sides seperated by the hyperplane , it’s always true that crosses at some point and we call that point and:
It is also true that:
Fact and contradicts each other, so the statement follows.
(b) There is a hyperplane that tolerates a noise radius
Let is the line connecting two points and , and , the hyperplane that satisfies the condition is orthogonal to and contains .