I am looking for a nice interpretation of the quantity
$M(x_1, \dots, x_N) = \max_{(i = 1, \dots, N)} c_i \cdot \min_{(i=1, \dots, N)} \frac{x_i}{c_i}$
where the $c_i$ are positive 'weights' and the $x_i$ are non-negative. The expression shares some properties of a mean, such as symmetry and homogeneity.
I am aware of Hölder means (https://en.wikipedia.org/wiki/Generalized_mean) and that Maximum and Minimum arise as limits for $p \to \pm\infty$, but even for the weighted Hölder mean the weights $c_i$ disappear as $p \to \pm \infty$.
Using the notation of Hölder means we have $$M_p(a_1,\ldots, a_n) := \left(\frac{1}{n} \sum_{k=1}^n a_k^p \right)^{1/p}$$ converges to $\max(a_1,\ldots, a_n)$ for $p \to \infty$ and to $\min(a_1,\ldots, a_n)$ for $p \to -\infty$. Hence, one can apply this to your expression by using $\max(a_1,\ldots, a_n) = (\min(1/a_1,\ldots, 1/a_n))^{-1}$:
$$M(x_1,\ldots, x_n)= \lim_{p \to \infty} \frac{M_p(c_1,\ldots, c_n)}{M_p(\frac{c_1}{x_1}, \ldots, \frac{c_n}{x_n})} = \lim_{p \to \infty} \left( \frac{\sum_{k=1}^n c_k^p }{\sum_{k=1}^n \left(\frac{c_k}{x_k}\right)^{p}} \right)^{1/p} $$
Maybe this helps a little bit with interpreting the expression, at least in terms of Hölder means.