condition for convex function to have unique minimum value

Question

The question is related to

Convex function with unique critical point is coercive

Assume we have a function $f: \mathbb{R}^{n} \to \mathbb{R}$ and we know that $f$ is just convex, not strictly nor necessarily strongly context. Also, $f$ is coercive.

Support is finite dimensional, but could be unbounded.

Is this enough condition for the function to have a unique minimum value?

Answer 1

I gather "coercive"—which is a term I was not previously acquainted with—simply means $\ \lim_\limits{\|x\|\rightarrow\infty}f(x)=\infty\ $. The function $\ f:\mathbb{R}^n\rightarrow\mathbb{R}\ $given by $$ f(x)\stackrel{\text{def}}{=}\cases{0&if $\ \|x\|\le1$\\ \|x\|^2-1&if $\ \|x\|\ge1$} $$ would then suggest to me that the answer to your question is "no".