Let a function $f$ on $[0, 1]^n$ be defined as $$f(x_1,\cdots, x_n)=\frac{1-\prod_i x_i} {\sum_i (1-x_i)}.$$

It is known that $1/n \le f(x_1, \cdots, x_n) \le 1$ and it is convex when $n=2$. Does the convexity hold for general $n$ and how to prove it?

  • $\begingroup$ Have you considered looking at tangent planes in $\mathbb{R}^{n-1}$. Does the function always lie above (in the gradient direction) the tangent plane? $\endgroup$ – user76844 Jan 21 '15 at 3:11
  • $\begingroup$ Yes. Though I don't see how that helps in the proof. $\endgroup$ – yml Jan 21 '15 at 20:38
  • $\begingroup$ Yes, as in you considered it, or yes, as in it's true? If you did consider it, what did you find? $\endgroup$ – user76844 Jan 22 '15 at 0:07

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