Perimeter Property of a Convex Polygon

The proof is based on drawing lines along the sides of the convex polygon, step-by-step cutting of the outer figure, and the triangle inequality principle.

Let us consider any side of the inner (convex) polygon and draw a line containing this side. This line will definitely intersect the enclosing (outer) polygon and cut off a certain part (a polygonal chain) that does not touch the inner figure.

According to the shortest path principle (the triangle inequality), the length of the segment between the endpoints of the cut-off polygonal chain is strictly less than the length of the cut-off polygonal chain itself. If we replace the cut-off polygonal chain with this new segment, we obtain a new polygon whose perimeter is less than the initial one and which still contains the inner convex polygon.

This process must be repeated step-by-step for each side of the inner polygon. Eventually, we will obtain a polygon that exactly coincides with the inner convex polygon. Since the perimeter was strictly decreasing at each step, we conclude that the perimeter of the initial outer polygon is necessarily greater than the perimeter of the inner figure.