1
$\begingroup$

Let $f:X \to \Bbb R$ be a convex function where $X \subseteq \Bbb R$ is a closed set. Does $f$ always attain a global minimum in $X$? If not, anyone can help give a counterexample?

$\endgroup$
  • 2
    $\begingroup$ Hint: there's a standard theorem about minimization of continuous functions on a closed set. $\endgroup$ – Brian Borchers Apr 5 '17 at 1:41
  • $\begingroup$ @BrianBorchers: This theorem is typically stated with compact sets. $\endgroup$ – gerw Apr 5 '17 at 14:44
2
$\begingroup$

Counterexample. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ defined by $f(x)=e^x$. We know that $\mathbb{R}$ is closed, but $f$ clearly doesn't have minumum.

$\endgroup$

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.