Let $(x\_1,...,x\_n)$ and $(y\_1,...,y\_n)$ be two different tuples of positive reals such that $x\_1\times\dots\times x\_n=y\_1\times\dots\times y\_n = c$.
Is it true that
$$\left(\frac{x\_1+y\_1}{2}\right)\times\cdots\times \left(\frac{x\_n+y\_n}{2}\right) > c?$$

I think this should follow from a concavity argument, perhaps on the function $f(x\_1,...,x\_n) = x\_1\times\cdots\times x\_n$, but not sure how exactly.