Given a vector $v \in \mathbb{R}^n$, is there a well known quantity equal or relating to $$ \frac{\max_j \vert{v_j}\vert}{\min_j \vert{v_j}\vert}? $$
I have noted that this quantity is coordinate-dependent (so it won't relate, e.g., to the norm of $v$), and that it's reminiscent of a condition number for matrices (though the similarity may be accidental).
Define $f(x)=1/x^2$ for $x>0$. Note that $f''(x)=6/x^4>0$. So $f$ is convex. Suppose we're going to choose one of $n$ random outcomes with equal probability $1/n$ and the value of the random variable $X$ in the case of outcome $i$ is $1/|v_i|$. By Jensen's inequality, $$\frac{1}{\Bigl(\frac{1}{n}\sum_{i=1}^n \frac{1}{|v_i|}\Bigr)^2} = f(\mathbb{E}[X])\leqslant \mathbb{E}[f(X)]=\frac{1}{n}\sum_{i=1}^n v_i^2=\frac{\|v\|^2}{n}.$$ Take square roots and multiply both sides by $\frac{1}{n}\sum_{i=1}^n \frac{1}{|v_i|}$ to get $$1\leqslant \frac{1}{n^{3/2}}\sum_{i=1}^n \frac{\|v\|}{|v_i|}.$$ Therefore $$n^{3/2}\|v\|\leqslant \sum_{i=1}^n \frac{\|v\|^2}{|v_i|}.$$ This lower bound is achieved with $\max_i |v_i|=\min_i |v_i|$. Indeed, if $|v_i|=\beta$ for all $i$, then $\|v\|=n^{1/2}\beta$ and $$\sum_{i=1}^n \frac{n\beta^2}{\beta}=n^2\beta = n^{3/2}\cdot n^{1/2}\beta=n^{3/2}\|v\|.$$