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Given $n<p \in \mathbb{N}$, I would like to show that the function $f_n: \mathbb{R}_+^p \rightarrow \mathbb{R}_+$ defined by $$f_n(x_1, x_2, \ldots, x_p) = \log \left( \sum_{1 \leq i_1 < i_2 < \ldots < i_n \leq p} a_{i_1, i_2, \ldots, i_p} \prod_{k=1}^n x_{i_k} \right)$$ is concave, where $a_{i_1, i_2, \ldots, i_p} \geq 0$ are weighing constants. Basically the function is taking the product of any $n$ out of $p$ non-negative variables and sum (with non-negative weights $a$) over all $\binom{p}{n}$ possible choices, and then taking the log.

I run the computer simulation and see the graph looks indeed concave, but I had a hard time proving it. Thai being said, it is also possible that this function is actually not concave, in which case a counter example would suffice. Any help is greatly appreciated!

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