On the Perimeters of Embedded Polygons
There are mathematical statements that appear counterintuitive. For example, when it comes to infinities, counterintuitive statements are abundant. At the other extreme, there are statements intuitively obvious that are rather hard to prove. Such, for example, is the famous Jordan Curve Theorem.
Naturally, mathematics does not lack in statements of any intermediate kind. Here is one that is intuitively obvious and admits several simple proofs, one of which (the one below) I judge rather elegant.
Proposition
The perimeter of a convex polygon situated wholly inside another polygon is less than the perimeter of the latter.
The perimeter of a convex polygon situated wholly inside another polygon is less than the perimeter of the latter.
Proof
Form semibands perpendicular to the sides of the inner polygon:
These semibands cut off the outer polygon pieces whose total length is not greater than its perimeter and is not less than the perimeter of the inner polygon.