Minkowski sum

DefinitionSuppose $A$ and $B$ are sets in a vector space $V$ over a field $K$,and suppose $\lambda \in K$. Then

$A+B$$=$$\mathrm{\{}a+b\mid a\in A,b\in B\},$

$A-B$$=$$\mathrm{\{}a-b\mid a\in A,b\in B\},$

$\lambda A$$=$$\mathrm{\{}\lambda a\mid a\in A\},$

$-A$$=$$(-1)A.$

The set $A+B$ is called the Minkowski sum of $A$ and $B$.If either $A$ or $B$ is a single point (a singleton), say $B=\{x\}$, then wewrite $A+x$ instead of $A+\{x\}$. Similarly we define $A-x$, $x-A$ and $x+A$.

Properties

Suppose $A$,$B$, $V$, and $\lambda$ are as above. Then

• $A+B=B+A$

• $\lambda (A+B)=\lambda A+\lambda B$

• $2A\subseteq A+A$, $3A\subseteq A+A+A$, etc,but in general, $A+A\ne 2A$.(Consider $A=\{(0,0),(0,1)\}$ in ${ℝ}^2$.)