Geodesic
On a minimax problem for ovals
Minimax theory and its applications, Tome 2 (2017) no. 2
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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
Mots-clés : Metric invariant, relative Chebyshev radius, isoperimetric inequality 
@article{MTA_2017_2_2_a4,
     author = {Rolf Walter},
     title = {On a minimax problem for ovals},
     journal = {Minimax theory and its applications},
     year = {2017},
     volume = {2},
     number = {2},
     zbl = {1380.51012},
     url = {http://geodesic.mathdoc.fr/item/MTA_2017_2_2_a4/}
}
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Rolf Walter. On a minimax problem for ovals. Minimax theory and its applications, Tome 2 (2017) no. 2. http://geodesic.mathdoc.fr/item/MTA_2017_2_2_a4/