Stability of the relative Chebyshev projection in polyhedral spaces

The paper is concerned with structural and stability properties of the set of Chebyshev centers of a set. Given a nonempty bounded subset $M$ of a metric space $(X,\varrho)$, the quantity $\operatorname{diam} M =\sup_{x,y\in M}\varrho(x,y)$ is called the diameter of $M$, and $r_M:=r(M):=\inf\bigl\{a\geqslant 0, \ x\in X \mid M\subset B(x,a)\bigr\}$, the Chebyshev radius of $M$. A point $x_0\in X$ for which $M\subset B(x_0,r(M))$ is called a Chebyshev center of $M$. The concept of a Chebyshev center and related stability, existence and uniqueness problems are important in various branches of mathematics. We study the structure of the set of Chebyshev centers and the stability of the Chebyshev projection (the Chebyshev center map). In the space $X=C(Q)$, where $Q$ is a normal topological space, we describe the structure of the Chebyshev center of sets with a unique Chebyshev center. The Chebyshev projection is the mapping associating with a nonempty bounded set the set of all its Chebyshev centers. Given a nonempty bounded set $M$ of a space $X$ and a nonempty set $Y\subset X$, the relative Chebyshev radius is defined as $ r_Y(M)=\inf_{y\in Y} r(y,M)$, where $ ~r(x,M):=\inf\bigl\{r\ge 0\mid M\subset B(x,r)\bigr\}=\sup_{y\in M}\|x-y\|$. The set of relative Chebyshev centers is defined as $ ~\mathrm{Z}_Y(M):=\{y\in Y\mid r(y,M)=r_Y(M)\}$. The mapping $M\mapsto \mathrm{Z}_Y(M)$ is called the relative Chebyshev projection (with respect to the set $Y$). Stability properties of the relative Chebyshev projection in finite-dimensional polyhedral spaces are studied. In particular, in a finite-dimensional polyhedral space, the projection $\mathrm{Z}_Y(\,\cdot\,)$, where $Y$ is a subspace, is shown to be globally Lipschitz continuous.