I'm given a function $g:\mathbb{R}^n \mapsto \mathbb{R}$, $g(y) = \prod_{i\in[n]} (1+y_i\cdot c_i)$, where $c_i>0$. Let $e_a,e_b$ be two arbitrary standard vectors. It is easy to show that for any $y,a,b$ the function $z \mapsto g(y + z(e_a - e_b))$ is concave.
My problem is the following. I need to consider, for a fixed $y$, a function of all possible swaps between standard vectors, i.e., $h\left( (z_{ab})_{a,b\in[n]} \right) = g(y + \sum_{a,b} z_{ab}(e_a-e_b))$. For such a function I would like to use Jensen's inequality, and so I need to argue that the function $h\left( (z_{ab})_{a,b\in[n]} \right)$ is concave. Can I argue as follows?
For any $(z_{ab})_{a,b\in[n]}$:
the function is strictly concave in every standard direction, i.e., $z_{ab}$, hence
since the function is $C^\infty$
in appropriately small neighborhood of $(z_{ab})_{a,b\in[n]}$ the function is concave, therefore
the function is concave everywhere.
