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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!

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