The mapping of compact into the set of its Chebyshev centres is   Lipschitz in the space l^n_{\infty}
  In this article the authors prove strong stability of the set of all Chebyshev centres of the bounded closed subset of the metric space. We endow the set of all compacts of the space $l^n_{\infty}$ with Hausdorff metric and prove that the map which puts in correspondence to each compact of $l^n_{\infty}$ the set of its Chebyshev centres is Lipshitz. 
Pyotr N. Ivanshin, Evgenii N. Sosov