Exercise 8.1

(a)

If there’s such a hyperplane that can tolerate noise radius greater than $\frac{1}{2}|x_{\text{+}}-x_{\text{−}}|$, we draw a line between two points, for $(x_{\text{+}},+1)$, we can pick a point on the line that just pass the middle point between $x_{\text{+}}$ and $x_{\text{−}}$ and still within the radius (which is greater than $\frac{1}{2}|x_{\text{+}}-x_{\text{−}}|$) of $x_{\text{+}}$. This point will have label $+1$. However, it’s obviously also in the radius of $x_{\text{−}}$, so it shall have a label of $-1$ as well. It is impossible to classify such point. This thus contradicts the fact that a hyperplane exists to tolerate such a noise radius. Our assumption is wrong, there’s no such hyperplane that can tolerate noise radius greater than $\frac{1}{2}|x_{\text{+}}-x_{\text{−}}|$.

(b)

We can choose the hyperplane that perpendicular to the line between $x_{\text{+}}$ and $x_{\text{−}}$ and passes through the middle point. The two balls with radius of $\frac{1}{2}|x_{\text{+}}-x_{\text{−}}|$ and centered at $x_{\text{+}},x_{\text{−}}$ are separated by this hyperplane totally. The projection of any point in the ball of $x_{\text{+}}$ on the norm of the hyperplane is positive.

Thus the hyperplane can tolerate such a noise radius.