Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I am trying to completely characterize the conditions on $f : \mathbb{R}^n \to \mathbb{R}$ under which $\{x | f(x) \le 0 \}$ is path-connected. There are many obvious conditions that are sufficient (e.g. $f$ concave), but is there any suite of conditions that is necessary and sufficient?

We can assume $f$ is continuous.

If you have any ideas or recommended reading, I'd love to hear about it. Thanks!

share|improve this question
3  
You won't get anything nice just by demanding a single sublevel set is path-connected--this condition is very weak. Maybe something interesting can be said if you demand that ALL of the sublevel sets are path-connected. –  Jack Huizenga Oct 10 '12 at 0:26
add comment

1 Answer

up vote 2 down vote accepted

If $f$ is $C^1$ and coercive, the existence of a non-path connected sub-level set implies the existence of a critical point, e.g. by the mountain pass theorem. If $f$ is $C^2$, the Hessian of this critical point has a "strong" Morse index $m _ * $ (number of negative eigenvalues) and a "weak" Morse index $m ^ * $ (number of non-positive eigenvalues) related by $m _ * \le1\le m ^ *$. Therefore, on the lines of Jack Huizenga's comment above, if $f$ is known to have no such critical point, beside being coercive and smooth, we may conclude that all sub-level sets are path connected.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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