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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.

Gheehyun Nahm
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  • It is okay to do this by computer; if it is quite accurate. However, the way I found to do this was not. – Gheehyun Nahm Feb 26 '15 at 15:00
  • There are systematic ways to explicitly calculate the convex hull of a space curve. May I ask you why are you interested in this question? – Paata Ivanishvili Jun 23 '17 at 18:59
  • @PaataIvanisvili Unfortunately, this is an outdated question, but at this time I was working on my high school research project (on classifying certain extremal inequalities), and to check whether my idea worked, I had to sort of figure out the convex hull of those two curves. – Gheehyun Nahm Jun 24 '17 at 18:36
  • Sorry that I did not mention it on this post, but I posted the same question on MO: https://mathoverflow.net/questions/198754/convex-hull-of-the-union-of-two-parameterized-curves-in-mathbbr3 – Gheehyun Nahm Jun 24 '17 at 18:37

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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.

Joonas Ilmavirta
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