The Minkowski sum 
of two sets 
and 
in a vector space is given
by 
.
The Minkowski sum of two disks centered at 
and 
with radii 
and 
, respectively, is given by the disk
centered at 
with radius 
. The Minkowski sum of two balls
is given similarly.
If 
and 
are polyhedra, then 
is a polyhedron and every
extreme point of 
is the sum of an extreme point
in 
and an extreme
point in 
.
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.
 ,  |  |
| regular tetrahedron and regular tetrahedron | cuboctahedron |
| cube and regular octahedron | small rhombicuboctahedron |
| regular dodecahedron and regular icosahedron | small rhombicosidodecahedron |
For semialgebraic sets 
, membership can be written as
 |
Quantifier elimination removes the existential quantifiers to give a semialgebraic set
description of 
(Trott 2006). If 
is a convex body and 
is a closed ball centered at
the origin, then 
is a parallel body. For
translated sets, 
iff 
, which expresses an intersection
test as membership in a Minkowski difference.
The Minkowski sum operation is implemented in the Wolfram Language as RegionDilation.
