I am currently revising the material in Luenberger's classic "optimization by Vector Space Methods" for an upcoming oral exam, and find myself stuck in section 5.13.
I am stuck on the following notion: let $K$ be a convex set in a real normed vector space, $X$. The support functional of a convex set $K \subset X$ is $$h(x^*) = \sup_{x\in K} \langle x, x^*\rangle$$ where $x^* \in X^*$, an element of the dual of $X$.
I am confused about two properties of the hyperplane $\{x: \langle x ,x^* \rangle =h(x^*)\}$ (1., 2. below)
Luenberger claims, on page 136, that if $K$ does not contain $0$, then $h(x)$ may be regarded as the distance between $0$ to the hyperplane $\{x: \langle x ,x^* \rangle =h(x^*)\}$ for any $x^*$ with dual norm $1$. Here $h$ is the support functional of the convex set $K$, as defined above. I do not see why this is the case.
Moreover, there is a theorem on minimum norm duality just following this discussion, in the proof of which it is claimed that $h(x^*) = \inf_{x \in B(0,\epsilon^*)} \langle x, x^*\rangle$ where $\epsilon^*$ is the supremum over all radii such that the ball of radius $r$ around the origin is seperated from $K$ by the hyperplane $\{x: \langle x ,x^* \rangle =h(x^*)\}$. This looks very similar to 1. above but I don't see it...
I believe these claims to be closely related but I cannot really get my head around either of them. I think part of my issue is that Luenberger is using mostly geometric language in his proofs, which I am not entirely comfortable with, without the supporting algebraic characterization. Any clarification would be appreciated!
It was actually just an "easy" relaxation trick once I figured out how to write out Luenberger's statements more analytically. Set
$$h(x^*) = \sup_{x\in K} \langle x,x^*\rangle$$
and letting $\epsilon^* = \sup \{r : h(x^*) \leq \langle x,x^* \rangle, \forall x\in B(0,r)\}$, it is clear that $h(x^*) = \inf_{x\in B(0,\epsilon^*)} \langle x,x^*\rangle$ simply by optimizing over all upper bounds on $h(x^*)$...
This explains 2. and 1. is essentially now a consequence of the proof (of Theorem 1, section 5.13) (which I now think I understand).