  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.
