Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a convex function. Denote $X^*$ as the set of minimizers of $f$ and assume $X^*$ is unbounded. Is it possible that $\|g_x\|$ is unbounded when $d(x,X^*)$ is bounded where $g_x\in\partial f(x)$. Here $\partial f(x)$ denotes the set of subgradient vectors of $f$ at $x$ and $d(x,X^*)$ denotes the distance between point $x$ and set $X^*$.

I know $X^*$ is a closed convex set and I am thinking whether it is true that $\|g_x\|$ is bounded given $d(x,X^*)=1$. It must be true if $X^*$ is bounded, but not sure when it is unbounded. Also, does it help if we further assume $f$ is differentiable, so that $\partial f(x)=\{\nabla f(x)\}$?

**Edit**: I realized that what I really want to ask is whether it is possible that $\|g_x\|$ is unbounded but $d(x,X^*)$ is bounded. It is not equivalent to the boundedness of the ratio of $\|g_x\|$ and $d(x,X^*)$.