a affine hyperplane in normed space

Question

I was troubled by the above proof,how to conclude that $x^\prime=\frac{u}{\phi(u)}$?

Answer 1

The graph is in the plane passing through the origin, $x, y$. The vertex is the origin. Because $C$ is a cone, the intersection of that plane with $\partial C$ is two rays from the origin. $x', y', u, w$ are all points on this plane because they are linear combinations of $x, y$. Because both $u$ and $x'$ were chosen to lie on $\partial C$, (and on the same side), they both lie on the same ray. Which means $x' = ku$ for some $k$.

Since $x' \in H$, $$\phi(x') = 1\\\phi(ku) = 1\\k\phi(u) = 1\\k = \frac 1{\phi(u)}$$ So $x' = \dfrac u{\phi(u)}$.