User:Zfeinst/sandbox

Minkowski Difference

Definition
Let $$X$$ be a vector space. Let $$A,B \subseteq X$$ then the Minkowski difference $$A -^. B$$ is given by
 * $$A -^. B := \{x \in X: B + x \subseteq A\} = \bigcap_{b \in B} (A - b)$$

Relation to Minkowski addition
The Minkowski difference is (in general) different then the Minkowski addition of a set and the negative of a set. That is, $$A - B = \{a - b: a \in A, b \in B\} \neq A -^. B$$. Though if $$B = \{b\}$$ the two concepts coincide.

Let $$X$$ be a vector space. Let $$A,B \subseteq X$$. Then
 * $$B + (A -^. B) \subseteq A$$