Let $Q\subseteq \mathbb{R}^K$ be a convex closed set.
I would like your help to show or find the right reference in the literature for the steps below where I use the support function of $Q$ to represent the condition $q\in Q$. My questions are highlighted with number [1],[2] below.
In what follows I denote by $h(Q,b)$ the support function of $Q$ evaluated at $b\in \mathbb{R}^K$. I denote by $\mathbb{B}^K$ the unit ball in $\mathbb{R}^K$, i.e., the set of $b\in \mathbb{R}^K$ such that $b^Tb\leq 1$. I denote by $\mathbb{S}^K$ the unit sphere in $\mathbb{R}^K$, i.e., the set of $b\in \mathbb{R}^K$ such that $b^Tb= 1$
$$ q\in Q $$ $$ \Updownarrow \text{[1] Which Theorem/property am I using here?} $$ $$ b^T q - h(Q,b)\leq 0 \text{ }\forall b \in \mathbb{R}^K $$ $$ \Updownarrow \text{Given that the above holds also for $b=0_{K\times 1}$} $$ $$ \max_{b\in \mathbb{R}^K}[b^T q - h(Q,b)]=0 $$ $$ \Updownarrow \text{By positive homogeneity of $h$, I divide both sides by $||q||$ and can focus only on $b$ s.t. $b'b=1$} $$ $$ \max_{b\in \mathbb{S}^K}[b^T q - h(Q,b)]=0 $$ $$ \Updownarrow \text{[2] Which Theorem/property am I using here?} $$ $$ \max_{b\in \mathbb{B}^K}[b^T q - h(Q,b)]=0 $$ $$ \Updownarrow \text{$Q$ closed implies $\sup=\max$} $$ $$ \max_{b\in \mathbb{B}^K}[b^T q - \max_{\tilde{q}\in Q} b^T\tilde{q}]=0 $$