Solution
= Solution
Suppose $u,v\in U_n$ both attain the <local minimum>[minimum] distance $d$. Apply (b) to $x-u$ and $x-v$:
$$
\|u-v\|^2+4\left\|x-\frac{u+v}2\right\|^2=2\|x-u\|^2+2\|x-v\|^2=4d^2.
$$
The midpoint belongs to $U_n$, so its squared distance is at least $d^2$. Thus $\|u-v\|^2\le0$, forcing $\boxed{u=v}$. Existence will also follow constructively in (d)(i).