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$$ \sqrt[w]{x_{1}^{w_{1}} x_{2}^{w_{2}} \cdots x_{n}^{w_{n}}} $$

I am new in convex optimization, so I am confused to prove the equation. Given nonnegative parameters $w_i > 0$, $i=1, \ldots, n$ and $w = w_1 + \ldots + w_n$. Prove that the following function is concave in nonnegative $x_1, \ldots, x_n$. Thank you for your answer and help!

jacksons
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  • The function, not the equation, is concave. – YCor Feb 09 '20 at 10:20
  • Note that you can divide each $w_i$ by $w$, making the expression just $$x_1^{w_1} x_2^{w_2} \cdots x_n^{w_n}$$ with condition $w_1 + \ldots + w_n = 1$. Now note that for each individual coordinate $x_i$, this is just $Ax_i^{w_i}$ for some $A$ constant with respect to $x_i$. – Paul Sinclair Feb 09 '20 at 17:14

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