Wheeling around Chebyshev centers and Jung constant in normed planes

Given a normed plane X , a subset K of X is non-centerable if its diameter is smaller than twice its Chebyshev radius. We prove that for any non-centerable set K , the only Chebyshev center is the circumcenter of three points of K , which is an interior point of the convex hull of K . We give a simple and direct proof that the Jung constant of X is 4/3 if and only if the unit sphere is a regular hexagon. We present some examples of normed planes whose Jung constant is equal to 2/√3 2 / 3 , as in the Ecuclidean case, in particular those whose unit sphere is invariant under rotations of angle π /6 π / 6 radians. We compute some estimates of the Jung constant for different ₚ ℓ p normed planes.