On a Minimax Problem for Ovals
Keywords:
Metric invariant, relative Chebyshev radius, isoperimetric inequality.Abstract
For a bounded metric space (X,d) we consider the quantity δ(X) := inf p∈X sup q∈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, the invariant δ(X) seems rather
untouched in the geometric literature. Here we discuss, for a plane convex curve X = Γ, an isoperimetric type inequality between δ(Γ) and the perimeter L(Γ), namely L(Γ) π δ(Γ). Though the most general case is open there are classes of curves where definitive versions of the inequality are possible, including a discussion of equality. For quadrilaterals there is a surprising occurrence of ‘magic kites’ as possible extremals. A finite algorithm for polygons is established, and numerous experiments with it yield strong support for a general validity of the inequality
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