Given two polygons and and two vectors and let and be the vector sum. That is is the set . Also let the Minkowski sum of two sets A and B be as follows . The following lemma is on page 6 of the paper “Translational Polygon Containment and Minimal Enclosure using Mathematical Programming” by V.J. Milenkovic and Karen Daniels
The lemma is as follows: The set is non-empty iff
To prove the lemma we note that iff for some and . It then follows that for some and Thus is a point in and since then is also a point in hence is in
One question you might have is “what is the use of the lemma?”. The answer is that does not intersect iff Thus given two translations of and we have an algorithm for deciding if the translated polygons overlap.