Minkowski sum

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

$\displaystyle A+B$	$\displaystyle=$	$\displaystyle\{a+b\mid a\in A,b\in B\},$
$\displaystyle A-B$	$\displaystyle=$	$\displaystyle\{a-b\mid a\in A,b\in B\},$
$\displaystyle\lambda A$	$\displaystyle=$	$\displaystyle\{\lambda a\mid a\in A\},$
$\displaystyle-A$	$\displaystyle=$	$\displaystyle(-1)A.$

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 $A-x$ , $x-A$ and $x+A$ .

Properties

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

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$A+B=B+A$
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$\lambda(A+B)=\lambda A+\lambda B$
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$2A\subseteq A+A$ , $3A\subseteq A+A+A$ , etc, but in general, $A+A\neq 2A$ . (Consider $A=\{(0,0),(0,1)\}$ in $\mathbb{R}^{2}$ .)

Title	Minkowski sum
Canonical name	MinkowskiSum
Date of creation	2013-03-22 15:16:22
Last modified on	2013-03-22 15:16:22
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	4
Author	matte (1858)
Entry type	Definition
Classification	msc 20-00
Classification	msc 15-00
Classification	msc 13-00
Classification	msc 16-00
Related topic	VectorSpace
Related topic	Sumset