I need help understanding the last part of this proof, the lemma they are refering too is just a lemma about convex sets that shows us that p is unique:
I do not see how H* separates C and z.
The equation for H* is:
$(z-p)^Tx=\frac{(z-p)^T(z+p)}{2}=\frac{|z|^2-|p|^2}{2}$
And the elements in C sattisfies:
$(z-p)^Tx \le (z-p)^Tp$.
Now from what I understand in order for it to strongly separate we must have.
$(z-p)^Tp < \frac{|z|^2-|p|^2}{2}<(z-p)^Tz$.
Is it correct that this is what we must have? Is this inequality obvious?