Calculating the Convex hull of a specific set in $\mathbb{R}^3$ I have to calculate the convex hull of $A=\{(-4k,k^2+2,2k^2-2k)|k\ge 2\}\cup \{(k,k^2/4-1,-k^2/4-k)|l\le -2\}$. I am aware with the Fenchel-Bunt theorem, so I just have to consider every (closed) triangles made by $a,b,c\in A$. I wanted to do this by hand, but it was hard. I tried to do this by computer; I made a program, but neither worked in the way I wanted to. My question is : is there a way to calculate this convex hull? I want to find the maximum value and the minimum value of $z$ satisfying $(x,y,z)\in \mathrm{con}(A)$ while $x,y$ is given.
 A: This is only a partial answer.
It seems unlikely to me that there would be a full answer without a horrible mess of calculations (and even such calculations might fail to produce a nice final result).
Let us consider the projection $(x,y,z)\mapsto(x,y)$.
The projection of the convex hull of $A$ is the same as the convex hull of the projection of $A$.
The projection of $A$ consists of two plane curves.
If we define $f:(-\infty,-2]\to\mathbb R$ by
$$
f(t)
=
\begin{cases}
\frac1{16}t^2+2, & t\leq-8 \\
-\frac23-\frac56t, & -8<t\leq-2,
\end{cases}
$$
then the convex hull of the projection of $A$ is
$$
\{(x,y)\in\mathbb R^2;x<-2,y\geq f(x)\}\cup\{(-2,1)\}.
$$
Therefore your question (finding minimum and maximum for $z$, given $(x,y)$) only makes if $(x,y)$ is in this set.
I suspect that finding an explicit form for the bounds for $z$ in terms of $x$ and $y$ can be very difficult.
It would be a more reasonable task to find some estimates for these upper and lower bounds, but that doesn't seem trivial either.
