The Minkowski sum A + B of two sets A and B in a vector space is given by {a + b:a element A, b element B}. The Minkowski sum of two disks centered at x_1 and x_2 with radii r_1 and r_2, respectively, is given by the disk centered at x_1 + x_2 with radius r_1 + r_2. The Minkowski sum of two balls is given similarly. If A and B are polyhedra, then A + B is a polyhedron and every extreme point of A + B is the sum of an extreme point in A and an extreme point in B. For example, taking the Minkowski sums of the following pairs of Platonic solids in dual position but with unit edge lengths give the following polyhedra.
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