Existence of function with specific properties

Question

Given a function $f:\mathbb{R}\to\mathbb{R}$ and a point $p_0 \in \mathbb{R}$ define the set $$B = \{g : \mathbb{R} \to \mathbb{R} \mid g(p_0) = f(p_0), g \text{ is convex}, g(x) \geq f(x) \text{ for all } x \in \mathbb{R} \}.$$

Is there a function $m:B\to \mathbb{R}$ such that $m(g_1) \geq m(g_2)$ iff $g_1(x) \geq g_2(x)$ for all $x \in \mathbb{R}$?

Answer 1

No, this is not possible if you exclude the trivial case $B \ne \emptyset$.

You can always find two functions $g_1, g_2 \in B$, such that neither $g_1(x) \le g_2(x)$ for all $x$ nor $g_2(x) \le g_1(x)$ for all $x$ hold.