Minkowski sum - Knowlpedia

Core idea

For subsets $A,B$ of a real or complex vector space $V$ , their Minkowski sum is

A+B=\{a+b:a\in A,\ b\in B\}.

Scalar multiplication is defined similarly by $tA=\{ta:a\in A\}$ .

Algebraic properties

Set addition is associative and commutative, and $t(A+B)=tA+tB$ . It is not a group operation on subsets: cancellation can fail, and most sets have no additive inverse under Minkowski addition.

Convexity and neighborhoods

If $A$ and $B$ are convex, then $A+B$ is convex. In a normed vector space, $A+B_r$ is the Minkowski thickening of $A$ by radius $r$ .

References

Rolf Schneider, Convex Bodies: The Brunn–Minkowski Theory, 2nd ed., Cambridge University Press, 2014. DOI record.