Let $f:\mathrm{\Omega}\subseteq {\mathbb{R}}^{n}\to {\mathbb{R}}_{\ge 0}$ be a continuous differentiable function over $\mathrm{\Omega}$. Suppose that the function $f$ is concave, and fix two points $\mathbf{x}=({x}_{1},\dots ,{x}_{n}),\mathbf{y}=({y}_{1},\dots ,{y}_{n})\in \mathrm{\Omega}$,$\mathbf{x}=({x}_{1},\dots ,{x}_{n}),\mathbf{y}=({y}_{1},\dots ,{y}_{n})\in \mathrm{\Omega}$.

If ${x}_{i}\le {y}_{i}$ for all $i=1,\dots ,n$ and $\mathrm{\Omega}={\mathbb{R}}^{n}$, does it hold $\parallel {\mathrm{\nabla}}_{\mathbf{x}}f\parallel \ge \parallel {\mathrm{\nabla}}_{\mathbf{y}}f\parallel $?

If ${x}_{i}\le {y}_{i}$ for all $i=1,\dots ,n$ and $\mathrm{\Omega}={\mathbb{R}}^{n}$, does it hold $\parallel {\mathrm{\nabla}}_{\mathbf{x}}f\parallel \ge \parallel {\mathrm{\nabla}}_{\mathbf{y}}f\parallel $?