Let $f : \mathbb{R}^n \to \mathbb{R}_∞$ be a function.
I want to prove that $f$ is convex over the line $L_{v,x_0}$ iff $\psi : \mathbb{R} \to \mathbb{R}_∞$
$\psi(t) := f (x_0 + tv)$, is convex over $I (x_0, v )$.
where $L_{v,x_0}:= \{x_0 + tv : t ∈ \mathbb{R}\}$, and $I(x_0,v):=\{t \in \mathbb{R} : x_0 + tv \in C\}$
Any help is very much appreciated