Does there exist a convex function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ such that $S:=\arg\min_{x\in\mathbb{R}^n}f(x)\neq\emptyset$ and $S$ is bounded, provided that $f$ is not coercive? Recall that $f$ is coercive if $f(x)\to\infty$ as $\|x\|\to\infty$.
From the assumption, we know $f$ is continuous but possibly not differentiable. Since $S\neq\emptyset$, we know $f^*:=\inf_{\mathbb{R}^n}f(x)>-\infty$ and there exists $x^*$ such that $f(x^*)=f^*$. However, it seems that if $f$ is not coercive, then somehow the set of minimizer will stretch to infinity. To give counterexample, I am thinking of whether it is possible for $f$ to asymptotically approaches $f^*$ but never attains it as $\|x\|\to\infty$?