Let $Y$ be a Banach space and equip $Y$ with the weak topology. Now, let $X$ be a closed, bounded, and convex subset of $Y$ and suppose that the relative (weak) topology on $X$ is metrizable with metric $d_X$. Let $x_1,\dots,x_n \in X$ for some natural $n>0$.

Let $\emptyset\neq Z\subseteq X$ be a compact subset of $X$. Then, does the map $$ x\mapsto \min_{1\leq i\leq n}\, d_X(x_i,x) $$ admit a measurable selection?

Let $Y$ be a Banach space and equip $Y$ with the weak topology. Now, let $X$ be a closed, bounded, and convex subset of $Y$ and suppose that the relative (weak) topology on $X$ is metrizable with metric $d_X$. Let $x_1,\dots,x_n \in X$ for some natural $n>0$.

Let $\emptyset\neq Z\subseteq X$ be a compact subset of $X$. Then, does the map $$ x\mapsto \min_{1\leq i\leq n}\, d_X(x_i,x) $$ admit a measurable selection? I.e.: Does there exist a measurable function $$ S\mapsto \{1,\dots,n\} \mbox{ s.t. } d_{X}(x_{S(x)},x)=\min_{i=1,\dots,n} d_{X}(x_i,x) $$ for all $x\in Z$ and $S$ is measurable as a function from $Z$ to $\{1,\dots,n\}$ (where the latter has the $\sigma$-algebra $2^{\{1,\dots,n\}}$?)

Let $Y$ be a Banach space and equip $Y$ with the weak topology. Now, let $X$ be a closed, bounded, and convex subset of $Y$ and suppose that the relative (weak) topology on $X$ is metrizable with metric $d_X$. Let $x_1,\dots,x_n \in X$ for some natural $n>0$.

Let $\emptyset\neq Z\subseteq X$ be a compact subset of $X$. Then, does the map $$ x\mapsto \min_{1\leq i\leq n}\, d_X(x_i,x) $$ admit a continuous selection?

Let $Y$ be a Banach space and equip $Y$ with the weak topology. Now, let $X$ be a closed, bounded, and convex subset of $Y$ and suppose that the relative (weak) topology on $X$ is metrizable with metric $d_X$. Let $x_1,\dots,x_n \in X$ for some natural $n>0$.

Let $\emptyset\neq Z\subseteq X$ be a compact subset of $X$. Then, does the map $$ x\mapsto \min_{1\leq i\leq n}\, d_X(x_i,x) $$ admit a measurable selection?