  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.
