0
$\begingroup$

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?

References

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?

$\endgroup$
  • $\begingroup$ 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
2
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.