The perimeter of rounded convex planar sets (Q2456234)

The aim of the paper is to determine the extreme values of \(\Delta _a(K)=\text{perimeter}(K)-\text{perimeter}(K')\), as \(K\) runs over all convex sets inscribed into a rectangle \(ABCD\) of sides \(a\) and \(1/a\), and \(K'\) is the image of \(K\) by the composition of two orthogonal linear transformations parallel to the edges of the rectangle which makes a unit square out of the rectangle. For values of \(a\) close to 1, the minimum is negative, which is somewhat surprising. The extremal sets are neatly described in geometrical erms. The proof is a mixture of reasonings involving methods from variational calculus, discrete geometry and of explicit computations performed by a computer program.