Show that $\prod x_i \ge 1$ is a convex set

Question

I'm trying to generalize this question to arbitrary $n$ dimensions, i.e. to show that $C = \{x\in \mathbb R^n_{++} : \prod_i^nx_i \ge 1 \}$ is convex.

At first I thought maybe through induction, but I reached a dead end there. Any suggestions?

Answer 1

The set can be expressed as $$\left\{ x \in \mathbb{R}_{++}^n : \sum_{i=1}^n \log x_i \geq 0 \right\},$$ which is convex as $g(x)=\sum_{i=1}^n \log x_i$ is a concave function.

Answer 2

By weighted AM GM, for any $(x_i), (y_i) \in C$ and $t \in (0,1)$, we have

$$\prod_{i=1}^n (tx_i + (1-t)y_i) \ge \prod_{i=1}^n x_i^t y_i^{1-t} = \left( \prod_{i=1}^n x_i\right)^t \left(\prod_{i=1}^n y_i\right)^{1-t} \ge 1^t 1^{1-t} = 1$$