I am wondering about the set
$$\{(x,y,z)\in\mathbb R^3\mid xyz\geq a\text{ and } x,y,z\geq 0\}$$
for some $a>0$ and whether it is convex or not. I have tried to show that the function $f(x,y,z)=xyz$ is convex over $\{(x,y,z)\in\mathbb R^3\mid x,y,z\geq 0\}$ by considering its Hessian matrix but I think the function is not convex as the Hessian seems to not be positive semi-definite.
I was wondering if there is maybe an example of two points such that any convex combination is not in the set (which I doubt). Otherwise, I would be interested in some hints on how to solve the problem.
This is the epigraph of the function $g:(0,\infty)\times (0,\infty)\to\Bbb R$, $g(x,y)=\frac a{xy}$. Therefore, it is convex if and only if $g$ is convex. The hessian of $g$ is $\nabla^2g(x,y)=\frac a{xy}\begin{pmatrix}\frac {2}{x^2}&\frac{1}{xy}\\ \frac{1}{xy}&\frac {2}{y^2}\end{pmatrix}$, which is positive definite by Sylvester's criterion. Therefore $g$ is convex, and so is your set.