Is there a case where the perimeter of a convex poly is greater than the perimeter of its bounding box?

Question

I know that the area of the polygon is always smaller than the area of its bounding box, but can it have a greater perimeter than the perimeter of its bounding box?

Answer 1

First of all, the area of the polygon may equal that of the bounding box, so I'd say “no larger” instead of “smaller”.

The answer to your question is “No”:
A convex polygon is the intersection of a finite number of half planes, with the additional requirement that the result is compact. So you can obtain every convex polygon by taking its bounding box and cutting away half planes. Each time you cut away a part, you take away two sides of a triangle and replace them by the third. Due to the triangle inequality, this will never increase the perimeter.

Answer 2

In general, if A and B are convex polygons and A is a subset of B, then the perimeter of A is less than or equal to the perimeter of B.