Can the Minkowski sum of two convex sets in the plane which are not polygons be a polygon?

Kamden Larson 2022-10-21 Answered
Minkowski sum of convex sets in the plane which are not polygons
Can the Minkowski sum of two convex sets in the plane which are not polygons be a polygon? Explicitly my convex set is of the form C = { ( x , y ) R 2 : x , y 0  ,  y + x 1  and  y 1 x }
I am interested in knowing if there is a convex set C' such that the Minkowski sum
C + C = P
where P = { ( x , y ) R 2 : x , y 0  and  1 / 2 x + y 1 } .
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Answers (1)

relatatt9
Answered 2022-10-22 Author has 12 answers
Step 1
The Minkowsky sum of two convex sets is a compact polygon only if the two summands are compact polygons.
Consider first the case that C,C′ are compact. A compact convex set is a polygon if and only if there are only finitely many points that can be obtained as intersection of a line with C. Let c C and a line such that C = { c }. A line parallel to then intersects C′ in a segment [ c 1 , c 2 ] of its boundary (where possibly c 1 = c 2 ). Then [ c + c 1 , c + c 2 ] is part of the boundary of P parallel to . Let the preceeding edge of P be [ a , c + c 1 ] and the next edge be [ c + c 2 , b ]. Then the lines parallel to these through c bound C. The exterior angle at c is at least as big as the smallest exterior angle of P. We conclude that there can be at most finitely many such points c. Thus C is a polygon. By the same argument, C′ is a polygon.
Step 2
The details of what happens when C,C′ are not assume closed (they must of course still be bounded) are left as an exercise. At least it is immediately clear that C ¯ and C ¯ are polygons.
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