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Minkowski sum

(Definition)

Definition Suppose and are sets in a vector space over a field , 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

is called the Minkowski sum of

and

. If either

is a single point (a singleton), say $B=\{x\}$ , then we write

instead of $A+\{x\}$ . Similarly we define

and

Suppose

, and $\lambda$ are as above. Then

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

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Cross-references: singleton, point, field, vector space
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This is version 1 of Minkowski sum, born on 2005-05-16.
Object id is 7060, canonical name is MinkowskiSum3.
Accessed 2350 times total.

Classification:

AMS MSC:	15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
	20-00 (Group theory and generalizations :: General reference works )
	13-00 (Commutative rings and algebras :: General reference works )
	16-00 (Associative rings and algebras :: General reference works )

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