On a Minimax Problem for Ovals

TL;DR

This paper investigates a geometric invariant called the relative Chebyshev radius for plane convex curves, establishing an isoperimetric inequality and exploring extremal shapes through approximation and conjectures.

Contribution

It introduces and analyzes an isoperimetric inequality involving the Chebyshev radius for convex curves, providing reduction techniques and conjectures for extremal shapes.

Findings

Smooth case reduces to polygonal case via approximation

Squares are not the extremal figures for the inequality

Conjectures proposed for general convex curves

Abstract

For a bounded metric space $ X $ one can consider the quantity $ \delta(X) := \text{inf\rule[-0.5ex]{0em}{1ex}}_{\,p\in X}\; \text{sup}_{q \in X} \; d(p,q) $. This purely metric invariant is known from approximation theory as the relative Chebyshev radius of $ X $ w.r.t. $ X $ itself. Despite its obvious meaning, $ \delta(X) $ seems rather untouched in the geometric literature. In this paper we discuss, for plane convex curves $ X $, an isoperimetric type inequality, relating $ \delta(X) $ to the perimeter of the curve. Due to the minimax character of $ \delta(X) $, its handling resists the usual principles for extremal questions. It will be shown that the smooth case of the inequality can be reduced to the polygonal case by approximation. However, for polygons, there is the additional problem of the high dimensionality of the set of vertices. So, in general, we only can offer conjectures. Definite solutions are possible for restricted classes of curves. Even for short polygons, there already arises a sort of `magic kites' which show that squares are definitely not the extremal figures.