Let $V:\mathbb{R}^d \to \mathbb{R} \in \mathcal{C}^1$ such that $V$ is convex and $V$ has a unique critical point.

Then $V$ is coercive.

This was an example given in one my lectures. But it sounds like one of this examples for which there might a counterexample due to non-formal statement...

Can you prove that is right? Or there is such a counter-example?


The similar questions window tells me that in fact For a convex function, does having a unique minimizer imply that it is coercive? is not true in general Hilbert space. What about finite-dimensional ones?

  • $\begingroup$ Your reference already contains a proof for the finite-dimensional situation... $\endgroup$ – gerw Apr 11 '18 at 6:37
  • $\begingroup$ @gerw let me read It with more attention $\endgroup$ – Javier Apr 11 '18 at 6:41

Yes, that statement is correct but the proof is not so easy. See Corollary 8.7.1 in Rockafellar's Convex Analysis.


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