The extreme polygons for the self Chebyshev radius of the boundary

TL;DR

This paper solves an extremal problem for convex quadrilaterals, proving that the minimal perimeter shape with a fixed self Chebyshev radius is a magic kite, confirming a conjecture by Rolf Walter.

Contribution

It provides a complete solution for the minimal perimeter quadrilateral with fixed self Chebyshev radius, specifically identifying the magic kite as the optimal shape.

Findings

The minimal perimeter quadrilateral with fixed self Chebyshev radius is a magic kite.

Confirmed Rolf Walter's conjecture regarding the shape.

Solved the extremal problem explicitly for quadrilaterals.

Abstract

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.