What is a convex set $S = \mathbb{R}_{++}^2$

Question

I worked on the following task:

The function $f(x) = f(x_1, x_2) = x_1^{1/2} \cdot x_2^{1/2}$ is concave on the convex set $S = \mathbb{R}_{++}^2$.

I could solve the task with the Hessian matrix (being semi-definitive, i.e. conacave). What I do not really understand is the side note, that the the set is convex. The way I understand it, it has to do, that we only allow for positive inputs for $x_1$ and $x_2$. But what is the connection to convexity?

Answer 1

Q: do you know what a convex set is?

Hint Let $\mathbf{x}=(x_1,x_2)\in\mathbb{R}^2$, and $\mathbf{y}=(y_1,y_2)\in\mathbb{R}^2$. Consider $\lambda \in [0,1]$, and:

$$\mathbf{z}(\lambda)=\lambda \mathbf{x} + (1-\lambda) \mathbf{y}$$

is $\mathbf{z}(\lambda) \in \mathbb{R}^2$ true?