The extreme polygons for the self Chebyshev radius of the boundary

The paper is devoted to some extremal problems for convex polygons on the Euclidean plane, related to the concept of self Chebyshev radius for the polygon boundary. We consider a general problem of minimization of the perimeter among all $n$-gons with a fixed self Chebyshev radius of the boundary. The main result of the paper is the complete solution of the mentioned problem for $n=4$: We proved that the quadrilateral of minimum perimeter is a so called magic kite, that verified the corresponding conjecture by Rolf Walter.