Characterizing points by their distance to the unit ball Let $x,y\in\mathbb{R}^n$. Assume that, for all $z$ in the unit ball, $|x-z|=|y-z|=d_z$. From this we can deduce that $|x| = |y|$ since $0$ is in the unit ball. How can we show that $x=y$? I think it must be true but I cannot show it easily.

You can use the fact that $\bar x=x/|x|$ is the unique best approximation of $x\in \mathbb{R}^n\setminus B$ in $B$.
Since
$$
|y-\bar x|=|x-\bar x|<|x-z|=|y-z|
$$
for $z\in B\setminus\{\bar x\}$ by assumption, $\bar x$ is also a best approximation of $y$ in $B$. By uniqueness it follows that $\bar x=\bar y$. As you already know that $|x|=|y|$, you are are done.
The case $x\in B$ is trivial since you only need to use $0=|x-x|=|y-x|$. Finally, if you work with the open instead of the closed unit ball, you can simply replace $B$ by a closed ball of smaller radius in the argument above.
