Proving that the maximizer of the product function in a convex set dominates the midpoint of that set

Question

Let $S\in\mathbb{R}^n$ be a convex set containing the zero vector $(0,...,0)$, and assume that for all $i=1,...,n$ $$\max_{x\in S\cap\mathbb{R}^{n}_{+}}x_{i}=1.$$ To prove is then that $$\arg\max_{x\in S\cap\mathbb{R}^{n}_{+}}\prod_{i=1}^{n}x_{i}\geq(1/n,...,1/n)$$ I suspect that the argument must involve the fact that $(1/n,...,1/n)$ is the maximizer of the product function in the unit simplex, to which we then apply a concave transformation, but this is just a guess. Any help would be much appreciated.