Extreme problems for convex curves with given relative Chebyshev radius

TL;DR

This paper investigates extremal problems for convex curves and polygons in the plane, focusing on the relative Chebyshev radius, including its calculation for triangles and bounds on perimeters for given radii.

Contribution

It determines the relative Chebyshev radius for triangles and establishes maximal perimeters for convex curves and polygons with a specified radius.

Findings

Calculated the relative Chebyshev radius for arbitrary triangles.

Derived the maximum perimeter for convex curves with a given radius.

Extended extremal problem solutions to convex polygons.

Abstract

The paper is devoted to some extremal problems for convex curves and polygons in the Euclidean plane referring to the relative Chebyshev radius. In particular, we determine the relative Chebyshev radius for an arbitrary triangle. Moreover, we derive the maximal possible perimeter for convex curves and convex n-gons of a given relative Chebyshev radius.