We know
$$\|x\|_2 \leq \|x\|_1 \leq \sqrt{d} \|x\|_2$$ for $x = (x_1, x_2,\dots, x_d)$. If we also have $x_i \geq 0$, is any way to know how tight this inequality can be? In other words, are there specific conditions (e.g., sparsity, high dimensionality) to get $\|x\|_2 - \|x\|_1 \leq \epsilon$ where $\epsilon$ is a very small positive number?