[ Triangle Inequality (miscellaneous).]
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.
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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