Efficient way of checking if a set is convex in 3 or more dimensions?

Question

Consider the following set: $$C:=\Bigg\{\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}: x_3=\lvert x_2\lvert,\:x_1\leq3\Bigg\}$$ Now, I can graph this using a program, like Mathematica, and tell visually whether or not its convex. However, I was wondering how to do this without relying on such method because if we were dealing with more than 3 dimensions then it would be hard or impossible to realize at that point. What's the best method?

Answer 1

The second constraint is linear.

Let's look closer to the first constraint.

$(0,1,1)$ is in $C$, $(0,-1, 1)$ is in $C$.

Consider $\frac12(0,1,1)+\frac12(0,-1,1)=(0,0,1)$ and it is not in $C$. Hence it can't be convex.