Let $f : \mathbb{R}^n → \mathbb{R}_∞$ be a function and let $C ⊂ dom f$ be a convex set. $$**Part I**$$ Prove that $f$ is a convex function if and only if $f$ is convex over every line $L_{v,x_0}$ where $L_{v,x_0} := \{x_0 + tv : t \in \mathbb{R}\}$.
i.e. f is convex iff when restricted to any line, it be convex on that certain line.
So essentially to handle this proof I need to show that convexity for $f$ holds when its convex over any arbitrary line, and on the converse it does not hold when $f$ is not convex over that arbitrary line. If $f$ is convex then, $$\forall x,y \in C, a \in [0,1]$$ it holds that $$f(ax+(1-a)y) \leq af(x) + (1-a)f(y) \space \space\space\space\space\space\space\space\space\space\space\space (1)$$
EDIT: $$**Part II**$$
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