I met problems about sublevel sets of proper convex functions.
Suppose $f: R^n \Rightarrow \overline{R}$ is a proper convex function, and $\alpha > \inf f$. Denote $S_{\alpha}$ the $\alpha$-sublevel set of $f$, i.e., $S_{\alpha} = \{ x: f(x) \le \alpha \}$. Consider another point $d$, which satisfies $dist(d, S_{\alpha}) \ge r > 0$, where $r > 0$ is an absolute constant. I think there may exist some absolute constant $t > 0$ so that $f(d) \ge \alpha + t$ holds for all such $d$. But, I don't know whether this is correct. If it is, how to prove this?
Thanks for any help!
I'm really not a convex analysis kind of guy, but what about $$f(x_1, ..., x_n) = x_1^4$$ If we fix $r$, we have that $$|f(x_1, 0 ,..., 0) - f(x_1+r, 0, ..., 0)| \to \infty$$
as $x_1 \to \infty$. Working backwards we can deduce that there is no such $t$ that works for all points in $\mathbb{R}^n$.