# minimum of some functions

Denote $U=\{(x_1,x_2,...,x_n):x_j\in(0,1) (1\le j\le n),\sum_{j=1}^nx_j=1\}$.

Let $f_i=f_i(x_1,x_2,...,x_n)$ ($1\leq i\leq n-1$) be $n-1$ real functions which satisfy:

$\prod_{i=1}^{n-1}f_i=\prod_{j=1}^nx_j$ and $\sum_{i=1}^{n-1}f_i=1-x_n-\sum_{j=1}^{n-1}x_j^2$ for any $(x_1,x_2,...,x_n)\in U$.

How to compute $\min_{1\leq i\le n-1}\min_{(x_1,x_2,...,x_n)\in U}\{f_i(x_1,x_2,...,x_n)\}$? Whether it equals to a strictly positive real number which is independent of $(x_1,x_2,...,x_n)$? Thank you so much.

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Also posted at MSE: math.stackexchange.com/questions/344796/… I put this link here solely so that people who might answer this question can check first if it has been answered or discussed on MSE. –  Yemon Choi Mar 28 '13 at 19:24