Suppose $C \subseteq \mathbb{R}^{2n}$ is a closed, bounded, convex set, with support function $h: \mathbb{R}^{2n} \rightarrow \mathbb{R}$, defined as $$h(c_1, c_2) := \sup \{c_1'x + c_2'y : (x,y) \in C\}~.$$
If all I know about $C$ is its support function (in a sense, this means I "know" $C$ itself), is it possible to obtain a reasonable expression for $$H_c(x) := \sup \left\{c'y: (x,y) \in C\right\}~,$$ for a known $c \in \mathbb{R}^n$, ideally in terms of the support function $h$? What if I also know the $\arg\sup$? (not sure what this function is called): $$s(c_1, c_2) := \arg\sup \{c_1'x + c_2'y : (x,y) \in C\}~.$$
I am especially interested in computing the quantity $\sup_{x \in C_x} \frac{H_c(x)}{c'x}$ (assume that $C$ bounds $x$ away from zero) where $C_x$ is the projection of $C$ onto it's first $n$ coordinates. We can, of course, obtain $\sup_{x \in C_x} H_c(x) = h(0,c)$, but it's not clear to me how to translate this to obtain $\sup_{x} \frac{H_c(x)}{c'x}$.