Let $f \colon \Omega \subseteq \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ be a continuous differentiable function over $\Omega$. Suppose that the function $f$ is concave, and fix two points $ \mathbf{x} = (x_1, \dots, x_n), \mathbf{y} = (y_1, \dots, y_n) \in \Omega $.
$(1) $ If $x_i \leq y_i$ for all $i = 1, \dots, n$ and $\Omega = \mathbb{R}^n $, does it hold $ \parallel \nabla_{\mathbf{x}} f \parallel \geq \parallel \nabla_{\mathbf{y}} f \parallel $?
$(2) $ If $f(\mathbf{x}) \leq f(\mathbf{y})$ and $\Omega = \mathbb{R}^n $, does it hold $ \parallel \nabla_{\mathbf{x}} f \parallel \geq \parallel \nabla_{\mathbf{y}} f \parallel $?
$(3) $ Does points $(1)$-$(2)$ hold, if we consider proper subsets $\Omega $ of $\mathbb{R}^n $?
partial answer: (1) is false. Take a concave function which is symmetric about the origin (e.g. $f=-\|x\|^2$). if $0\leq x_i$ and $\|\nabla f(\mathbf{x})\|>\|\nabla f(0)\|$, then due to symmetry we'd get $-x_i\leq 0$ but $\|\nabla f(-\mathbf{x})\|<\|\nabla f(0)\|$.