10
$\begingroup$

I'm trying to determine the conditions on $f : \mathbb{R}^n_{\ge 0} \to \mathbb{R^n}$ under which $\{ x | f(x) \le 0\}$ is path-connected. We can assume that $f$ is continuous and concave (i.e. for any $\lambda \in [0, 1]$, $f(\lambda x + (1 - \lambda) y) \ge \lambda f(x) + (1 - \lambda) f(y)$).

Inequalities on $\mathbb{R}^n$ are pointwise: $a \ge b$ iff $a_i \ge b_i$ for each $i$.

Thanks!

$\endgroup$
0
$\begingroup$

Note $L_{\lambda} = \{x\in\mathbf{R}^n\;|\;f(x)\leq \lambda\}$. Asking path-connectedness of only $L_0$ is way too weak. If $f$ is of class $\mathscr{C}^1$ and is coercive then if there is a $\lambda$ such that $L_{\lambda}$ is non path-connected implied the existence of a critical point, this is the so-called "moutain pass theorem". (See for instance the nice book of Aubin and Ekeland, "Applied non-linear analysis".) If $f$ is of class $\mathscr{C}^2$, the Hessian operator of $f$ at this critical point will have at least one positive eigenvalue and at most one negative eigenvalue. (See weak and strong Morse indexes theory for instance.) So if $f$ hasn't such critical point, and is smooth enough, all $L_{\lambda}$ will be path-connected.

$\endgroup$

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.