Let $C \subseteq \mathbb R_+^2$ be the Pareto front of a pair of nonnegative functions $(f,g)$ (where $f$ is strongly convex and $g$ is convex, for simplicity), i.e,
$$ C := \{f(x(t)),g(x(t)) \mid t \ge 0\}, $$
where $x(t)$ is a minimizer of $L_t(x) := t f(x) + g(x)$. For any $\epsilon \in \text{range}(f)$, let $s(\epsilon) := \inf_{f(x) \le \epsilon} g(x)$. It is clear that $(\epsilon,s(\epsilon)) \in C$.
Question 1. Is there technical name for the function $s$ ?
Question 2. Is it true that $C$ equals the graph of $s$, i.e is it true that $C = \{(\epsilon,s(\epsilon)) \mid \epsilon \ge 0\}$ ?