# Minkowski sum and product of 2D-regions

Let $A_j$, $j=1,\dots,n$ be subsets of $\mathbb C$ defined by $$A_j=\{z\,|\,f_j(z)\geq0\}$$ where $f_j:\mathbb C\to\mathbb R$ are given functions.

I am interested in the following subsets of $\mathbb C$ (maybe they could be called Minkowski sum and product respectiverly): $$A_1+A_2+\dots+A_n := \{z_1+z_2+\dots+z_n\,|\,z_j\in A_j\,\forall j=1,\dots,n\} \,,$$ $$A_1\cdot A_2\cdot\dots\cdot A_n := \{z_1 z_2\cdots z_n\,|\,z_j\in A_j\,\forall j=1,\dots,n\} \,.$$ How can I plot them in Mathematica?

Example Let's take the ball $$A_1=A_2=\{z\in\mathbb C\,|\, |z-1|\leq 1\} = \{1+r\,e^{it}\,|\,0\leq r\leq1,\,0\leq t<2\pi\}$$ then I would like to plot the region in $\mathbb C$ obtained by multiplying any two points in the ball, that is $$A_1\cdot A_2=\{z_1\cdot z_2 \,|\, |z_1-1|\leq 1,\, |z_2-1|\leq1\} =\{(1+r_1e^{it_1})(1+r_2e^{it_2}) \,|\, 0\leq r1,r_2\leq1,\,0\leq t_1,t_2<2\pi\} \,.$$ But I am not able to do it because ParametricPlot requires only 2 parameters, while I have 4 (that are $t_1,t_2,r_1,r_2$).

• I don't know, since ParametricPlot seems to accept only two paramters for a 2dimensional region.. Jul 9 '15 at 16:56
• I am not sure that you provided enough detail for us to help you effectively. What are the $f$ functions? In general, even in an enclosed region, shouldn't there be an infinite amount of points $z$ that satisfy the $f(z)>0$ condition? It is not clear to me, then, how exactly I should interpret your sum in such a case. Jul 9 '15 at 17:18
• @MarcoB I added an example to be more clear. My sets are infinite indeed. Jul 9 '15 at 18:18
• This works on reals but not on complex numbers: Reduce[Exists[{x1, x2}, x1 ∈ Reals && x2 ∈ Reals && Abs[x1 - 1] <= 1 && Abs[x2 - 1] <= 1 && x1 x2 == x], x, Reals] gives 0 ≤ x ≤ 4.
– user484
Jul 9 '15 at 20:06

If

1. you have only two regions $A_1$ and $A_2$, and
2. every point in their sum/product can be attained as the sum/product of points on the boundaries of $A_1$ and $A_2$,

then you could do something like this:

ParametricPlot[
Through@{Re, Im}@((1 + Exp[I t1]) (1 + Exp[I t2])),
{t1, -Pi, Pi}, {t2, -Pi, Pi}, Mesh -> 7] 