How do I justify that a second order cone is an intersection of half space

Question

I am studying convex optimization right now, and the text book claims that a second order cone is a collection of intersections of half space

$$K_n = \bigcap_{ u:\|u\|_2 \leq 1} \{(x,y) \in R^{n+1 } \::\:\: y \geq u^Tx\}$$

I am interested in seeing a proof for the statement, how would I justify the claim?

Answer 1

The key to your formula is $$\|x\| = \sup_{\|u\|_2\le1} u^\top x.$$ This should help showing the equality above.

Answer 2

gerw is right.

$u^Tx\leq \|u \|_2\|x\|_2$, restricting that $\|u\|_2 \leq 1$, then $u^Tx\leq \|u \|_2\|x\|_2\leq \|x\|_2\leq y$.