Centers and nearest points of sets

Abstract:

For a Banach space $X$ and a subset $A$ of $X$, ${c_A}$ denotes the Čebyšev center of $A$ and ${P_A}x$ denotes the nearest point in $A$ to the point $x$ in $X$. The space of all subsets of $X$ is furnished with the Hausdorff metric. The modulus of continuity of the function $A \to {c_A}$ is computed in the case when $X$ is a Hilbert space and the sets $A$ are compact; the same is done for the function $A \to {P_A}x$, for fixed $x$, in the case when $X$ is uniformly convex and the sets $A$ are convex and closed.