Let C = $ \{x \in \mathbb{R}^n | \Pi x_i \geq 1\} $. The goal is to show that this is convex.
I have tried to use the definition with looking at the line segment formed by two vectors in C, but I was unable to conclude that the segment was in C. Instead, I am trying to find a convex set for which C is a sub-level set, but I am having no luck there either because I can't find an example of a convex set for which C would "fit in"
This is the same as
$$ \left\{ x \in \mathbb{R}^n \mid \sum_{i=1}^n \log(x_i) \geq 0 \right\} = \left\{ x \in \mathbb{R}^n \mid - \sum_{i=1}^n \log(x_i) \leq 0 \right\}, $$
the sublevel set of the convex function $-\sum_{i=1}^n \log(x_i)$ (since $\log$ is concave).