Minkowski sum
Definition Suppose $A$ and $B$ are sets in a vector space^{} $V$ over a field $K$, and suppose $\lambda \in K$. Then
$A+B$  $=$  $\mathrm{\{}a+b\mid a\in A,b\in B\},$  
$AB$  $=$  $\mathrm{\{}ab\mid a\in A,b\in B\},$  
$\lambda A$  $=$  $\mathrm{\{}\lambda a\mid a\in A\},$  
$A$  $=$  $(1)A.$ 
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 we write $A+x$ instead of $A+\{x\}$. Similarly we define $Ax$, $xA$ 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 ${\mathbb{R}}^{2}$.)
Title  Minkowski sum 

Canonical name  MinkowskiSum 
Date of creation  20130322 15:16:22 
Last modified on  20130322 15:16:22 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  4 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 2000 
Classification  msc 1500 
Classification  msc 1300 
Classification  msc 1600 
Related topic  VectorSpace 
Related topic  Sumset 