a) Prove that when transitioning from a non-convex polygon to its convex hull, the perimeter decreases. (The convex hull of a polygon is the smallest convex polygon containing it.) b) Inside a convex polygon, there lies another convex polygon. Prove that the perimeter of the outer polygon is not less than the perimeter of the inner one.
Problem 1171
Official solution
a) When transitioning from a non-convex polygon to its convex hull, some broken lines formed by the sides are replaced by straight-line segments (see figure). It remains to note that the length of the broken line is greater than the length of the segment with the same endpoints.
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b) Construct half-strips on the sides of the inner polygon, facing outward; the parallel edges of the half-strips are perpendicular to the corresponding side of the polygon (see figure). Let be the part of the perimeter of the outer polygon that lies within these half-strips. Then the perimeter of the inner polygon does not exceed , while the perimeter of the outer polygon is greater than .
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