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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 $p \to \pm \infty$.

MKR
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  • The $c_i$'s are just rescaling the variables, so where's the mystery ? (Or maybe can't I decode your formula.) –  Jun 30 '15 at 08:57
  • @yves-daoust: Yes, but they appear also in the factor $\max_{i=1, \dotsc, N} c_i$, so it is a little bit more than rescaling. From a mathematical point of view the formula is very simple of course, but I was wondering whether this type of expression appears elsewhere. – MKR Jun 30 '15 at 09:15
  • This is a constant factor, just a rescaling of the result. You can rewrite the weights as normalized ones, $c_i/c_{max}$. –  Jun 30 '15 at 09:21

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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.

j4GGy
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