Extremal Problems for Convex Curves with a Given Self Chebyshev Radius

The extreme polygons for the self Chebyshev radius of the boundary

“…In particular, the following result given in [19] holds: For each triangle P in the Euclidean plane, one has L(Γ) ≥ 2 √ 3 • δ(Γ) with equality exactly for equilateral triangles, where Γ is the boundary of P . The proof of this result was simplified in [8], where the authors determined the self Chebyshev radius for the boundary of an arbitrary triangle. Moreover, some related problems were considered in detail in [8].…”

“…The proof of this result was simplified in [8], where the authors determined the self Chebyshev radius for the boundary of an arbitrary triangle. Moreover, some related problems were considered in detail in [8]. In particular, the maximal possible perimeter for convex curves and boundaries of convex n-gons with a given self Chebyshev radius were found.…”

“…As the authors of [8] discovered somewhat later, the original problem from [19], which was to prove the inequality L(Γ) ≥ π • δ(Γ) for any closed convex curve Γ in the Euclidean plane, had already been solved many years ago in paper [11] by K.J. Falconer (the results of that paper were formulated in other terms, without using the self Chebyshev radius).…”

“…Recall that we use the term "extremal" in this paper only for polygons with minimal perimeter among all polygons with a given self Chebyshev radius of the boundary. It should be noted that the polygon with maximal perimeter among all polygons with a given self Chebyshev radius of the boundary was determined in [8,Theorem 4]. The boundary of such a polygon has only one self Chebyshev center.…”