Minkowski addition

In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors A and B in Euclidean space is formed by adding each vector in A to each vector in B, i.e., the set

Analogously, the Minkowski difference (or geometric difference) is defined as

It is important to note that in general ${\displaystyle A-B\neq A+(-B)}$. For instance, in a one-dimensional case ${\displaystyle A=[-2,2]}$ and ${\displaystyle B=[-1,1]}$ the Minkowski difference ${\displaystyle A-B=[-1,1]}$, whereas ${\displaystyle A+(-B)=A+B=[-3,3].}$ The correct formula connecting Minkowski sum and difference is as follows (here ${\displaystyle X^{c}}$ stands for the complement of ${\displaystyle X}$):

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