How do I do the Minkowski sum of two sets?

Question

I have 2 sets, $X$ and $Y$. How do I find the Minkowski sum of these sets?

Answer 1

Another way is with Tuples and Total.

A = {1, 2, 3};
B = {a, b, c};

MinkowskiSum[lis__] := Total[Tuples[{lis}], {2}]

MinkowskiSum[A, B]

{a+1, b+1, c+1, a+2, b+2, c+2, a+3, b+3, c+3}

A1 = {1, 2};
A2 = {a, b};
A3 = {x, y};

MinkowskiSum[A1, A2, A3]

{a+x+1, a+y+1, b+x+1, b+y+1, a+x+2, a+y+2, b+x+2, b+y+2}

Answer 2

For example

x = {{1, 0}, {0, 1}, {0, -1}};
y = {{0, 0}, {1, 1}, {1, -1}};

Union @@ Outer[Plus, x, y, 1]

{{0, -1}, {0, 1}, {1, -2}, {1, 0}, {1, 2}, {2, -1}, {2, 1}}

---

You can use CirclePlus (⊕) (entered as \[CirclePlus] or Escc+Esc) to define your Minkowski sum:

CirclePlus[x__] := Flatten[Union @@ Outer[Plus, x, 1], Length@{x} - 2]

x⊕y

{{0, -1}, {0, 1}, {1, -2}, {1, 0}, {1, 2}, {2, -1}, {2, 1}}

{{1,2}}⊕{{a,b}}⊕{{c,d}}

{{1 + a + c, 2 + b + d}}