Find $C\subset \mathbb{R}^2$ convex unbounded such that $\vert C \vert $ is not convex?

Question

The question is almost posted in the title and one thing to put is that

$$\vert C \vert : = \big\{ ( \vert x \vert , \vert y \vert )\in [0, +\infty)^2 \, : \,\, (x, y)\in C \,\, \big\} $$

If $C$ is bounded, then it is quite easy to construct an example in which $\vert C \vert $ is not convex. I appreciate it if someone can provide some idea to attack the unbounded case.

Answer 1

Notice that if $x$ does not change sign but $y$ does then $\lvert C\rvert$ folds the space across the $x$-axis. So to construct the set you seek find an unbounded set that is convex which when folded loses convexity due to the crease created. To get a picture of what I mean look at the graph of $y=x$ and the graph of $y=\lvert x\rvert$. The original graph was convex but when folded across the axis for $x<0$ a crease is formed that prevents convexity. This example does not quite work in your case but maybe a modification of this idea can.

An example of the set you seek is given below.

Consider the set $C=\{(x,y):y=x-1,x\ge0\}$. Then $C$ is unbounded and convex but $\lvert C\rvert$ is not convex.