Perimeter of rounded convex planar sets
Perimeter of rounded convex planar sets
L. Csirmaz:
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.