This question is related to [How to make a sandwich from just one piece of bread?](https://mathoverflow.net/questions/262919), asked on Feb 23 '17 by erz, and it goes as follows: 



> 
> **Question.** If a convex closed and bounded region $C$ in the plane $\mathbb{R}^2$ can be cut along some straight line into two congruent
>  pieces, must $C$ have either axial or central symmetry?
> 
> 
> 


To be specific about what $cutting$ means, a $piece$ consists of the closure of the set of all points of $C$ that lie on the same side of the cutting line. Obviously, if $C$ has either axial or central symmetry, then it can be cut so.


The question can be phrased generally in $\mathbb{R}^n$, where we would cut a convex body along a hyperplane and consider $n$ kinds of symmetry, for example central, axial, and mirror in $\mathbb{R}^3$. I believe the answer is $yes$. Is it perhaps known already?


In retrospect (see Wlod AA's very nice answer): my intuition was way off.



Cutting a convex body into two congruent pieces
Here is a counter-example (in the complex plane $\mathbb C=\mathbb R^2.)\ $ Let


$$\ P\ :=
\ \{ (x\ y)\in\mathbb C : 0\le x\le 1\quad\&\quad 0\le y\le 1-x^2\} $$


Then,


$$ C\,\ :=\,\ P\,\cup\, i\!\cdot\! P $$


The imaginary line is the requested cut.


