Minkowski addition

Remarks

Example: in $\R^2$, let $A = [0,1] \times \set{0}$ and $B = \set{0} \times [0, 1]$. A point $(s, 0) + (0, t)$ with $s, t \in [0, 1]$ is $(s, t)$, so $A + B$ is the unit square $[0,1]^2$.

Minkowski addition is associative and commutative, with neutral element $\set{0}$, and $A + \emptyset = \emptyset$. If $A$ and $B$ are convex, then $A + B$ is convex; if they are compact, then $A + B$ is compact.

Non-example of a familiar rule: $A + A = 2A$ can fail. For $A = \set{0, 1} \subseteq \R$ one has $A + A = \set{0, 1, 2}$ but $2A = \set{0, 2}$. For convex $A$ and $\lambda, \mu \ge 0$ one has $\lambda A + \mu A = (\lambda + \mu) A$. In terms of support functions, $h_{A + B} = h_A + h_B$ for nonempty compact convex $A, B \subseteq \R^n$. For polytopes, see the Minkowski sum of polytopes.