Suppose that $U \subseteq \mathbb{R}^d$, and satsifies

  • $U$ is dense in $\mathbb{R}^d$,
  • U has empty interior,

Then is it possible that $$ \inf_{x \in U} f(x) >\inf_{x \in \mathbb{R}^d} f(x), $$ for some (fixed) lsc function $f:\mathbb{R}^d\rightarrow \mathbb{R}$?


I'm always afraid of confusing lsc and usc, but what about $d=1$ and $U=\mathbb R\setminus\mathbb Q$ and $f(0)=0$ and $f(x) =1$ for $x\neq 0$?

  • 1
    $\begingroup$ Yes this is lsc. $\endgroup$ – YCor Feb 8 at 18:14
  • $\begingroup$ Cool, I came up with a slightly reduced version of this concentrated at a single point in $\mathbb{R}^d-U$. Thanks :) $\endgroup$ – AIM_BLB Feb 8 at 18:26

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

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