I'm having trouble with this one. Doing some research I find that some variable of interest depends on $$B=\left[\prod_{i=1}^{n}s_{i}^{s_{i}}\right]^{-1}$$ where $0<s_i<1$ for each $i$, $i=1\ldots n$ and $\sum_{i=1}^{n}s_{i}=1$.

To situate: imagine a stick of length 1 which is divided into $n$ shares denoted $s_i$. There is no relationship among the $s_i$ except $\sum_{i=1}^{n}s_{i}=1$

I want to study and say everything there is to say about $B$. Namely I have two questions at the moment:

  1. is $B$ of any recognizable form? It looks like (the inverse of) a weighted geometric average of the shares $s_i$, but with weights equal to the shares themselves, which makes it strange to me.
  2. Find the maximum of $B$. By numerical simulation it seems that $B$ reaches a maximum for “equitable shares” as in $s_i=(\frac{1}{2},\frac{1}{2})$ for $n=2$ shares ... $s_i=(\frac{1}{n},\dots\frac{1}{n})$ for $n$ shares. The maximum is then reached at $B=n$. (equivalently: it seems that the maximum of $B$ is reached for the arithmetic mean shares).

Any insights greatly appreciated. I'm more than rusty.

  • 3
    $\begingroup$ Notice that $\log B$ is the entropy of the probability distribution $s = (s_1, \cdots, s_n)$. It is well-known that it achieves the maximum at the uniform distribution $s=(\frac{1}{n},\cdots,\frac{1}{n})$, and this can be easily shown from the method of Lagrange multiplier together with strict concavity of the map $s\mapsto\log B$. $\endgroup$ Jun 28 '19 at 23:34
  • $\begingroup$ I suggest you look at the min of $ B^{-1}$ which is achieved at the same point as the min of $\sum s_i\log s_i$ since $\log$ is increasing. Observe that $r\to r\log r$ is convex on $[0,1]$ (taking it equal zero at zero for continuity) and thus the sum $\sum s_i\log s_i$ is convex as well. $\endgroup$
    – GReyes
    Jun 28 '19 at 23:41

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