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 $2^{\{1,\dots,n\}}$?)



Measurable selection for argmin to distance
Here is a simple direct argument. For $i=1,\ldots,n$, let $$C\_i=\{z\in Z\mid d(z,x\_i)\leq d(z,x\_j), j=1,\ldots,n\}.$$
Clearly, each $C\_i$ is closed and hence measurable. Let $M\_i=C\_i\setminus\bigcup\_{l=1}^{i-1}C\_l$. The nonempty sets of the form $M\_i$ form a finite measurable partition of $Z$. Now let $S$ map each $M\_i$ to $x\_i$.


