I need your help in solving the following problem:
Given a convex function $f : \mathbb{R}^n \to [0,\infty)$, let $c \in \mathbb{R}_+$ and let $L \subseteq \mathbb{R}^n$ be a non empty sublevel set of $f$ such that:
$$ L = \left\lbrace x \in \mathbb{R}^n \mid f(x) \leq c\right\rbrace$$
How can I prove that $\forall w \in \mathbb{R}^n$ such that $\left\| w \right\|_2 = 1$ there exists a real number $a > 0$ such that $$ aw \in L$$
Is it even true?
Please advise.
Not true. Let $f(x)=\sum_i (x_i-1)^2$, and $c=0$. Unless $w$ is such that $w_i=w_j$ for all $j$, $aw$ doesn't go through $L=\{1,1,\dots,1\}$.