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.