Minkowski sum of convex sets in the plane which are not polygonsCan 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 ′ = Pwhere P = { ( x , y ) ∈ R 2 : x , y ≥ 0 and 1 / 2 ≤ x + y ≤ 1 } .

Question

Minkowski sum of convex sets in the plane which are not polygonsCan 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&#039; such that the Minkowski sum  C  +      C    ′    =  Pwhere   P  =  {  (  x  ,  y  )  ∈            R        2    :  x  ,  y  ≥  0   and   1      /    2  ≤  x  +  y  ≤  1  }  .

relatatt9 · Accepted Answer

Step 1The 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   ℓ a line such that   ℓ  ∩  C  =  {  c  }. A line parallel to   ℓ 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   ℓ. 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 2The 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.

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

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