Existence of a Lipschitz selection of the Chebyshev-centre map

The paper is concerned with the existence of a Lipschitz selection for the operator $T_C$ (the Chebyshev-centre map) that assigns to any bounded subset $M$ of a Banach space $X$ the set $T_C(M)$ of its Chebyshev centres. It is proved that if the unit sphere $S(X)$ of $X$ has an exposed smooth point, then $T_C$ does not have a Lipschitz selection. It is also proved that if $X$ is finite dimensional the operator $T_C$ has a Lipschitz selection if and only if $X$ is polyhedral. The operator $T_C$ is also shown to have a Lipschitz selection in the space $\mathbf c_0(K)$ and $\mathbf c$-spaces.
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