On the Chebyshev Center and the Nonemptiness of the Intersection of Nested Sets

We show that if every bounded set in a Banach space has a Chebyshev center, then the intersection of nested closed bounded sets in this space is nonempty in the case of a critical parameter value. This result generalizes previously obtained sufficient conditions for the nonemptiness of the intersection in the critical case. We also answer a question posed by G. Z. Chelidze and P. L. Papini for Banach spaces satisfying the Opial condition for the weak-$*$ topology.