# 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

• $A+B=B+A$

• $\lambda(A+B)=\lambda A+\lambda B$

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

Title Minkowski sum MinkowskiSum 2013-03-22 15:16:22 2013-03-22 15:16:22 matte (1858) matte (1858) 4 matte (1858) Definition msc 20-00 msc 15-00 msc 13-00 msc 16-00 VectorSpace Sumset