The perimeter of rounded convex planar sets

A convex set is inscribed into a rectangle with sides a and 1/a so that the convex set has points on all four sides of the rectangle. By "rounding" we mean the composition of two orthogonal linear transformations parallel to the edges of the rectangle, which makes a unit square out of the rectangle. The transformation also applied to the convex set, which now has the same area, and is inscribed into a square. One would expect this transformation to decrease the perimeter. Interestingly this is not always the case. For each a we determine the largest and smallest possible increase of the perimeter. We also look at the case when the inscribed convex set is a triangle.