Originally Posted by

**Guy** I'm doing this problem as part of my effort to self-study my way through the first 7 chapters of Rudin.

Suppose k$\displaystyle \ge 3, x, y \in \mathbb{R}^k, ||x-y|| = d > 0$, and $\displaystyle r > 0$. Prove (a) that if $\displaystyle 2r > d$, there are infinitely many $\displaystyle z \in \mathbb{R}^k$ such that $\displaystyle ||z - x|| = ||z - y|| = r$, (b) there is exactly one such $\displaystyle z$ if $\displaystyle 2r = d$ and (c) there is no such z if $\displaystyle 2r < d$.

My progress:

I proved (b) and (c) using the condition for attainment in Cauchy-Schwarz. Part (a) I proved after assuming that $\displaystyle x = 0$ and $\displaystyle y = (y_1, 0, ..., 0)$. I'm probably just being stupid, but the problem I seem to be running into is that I'm having a hard time undoing my rotation and translation without going beyond what can be done with the material learned so far. Any help would be appreciated. If anyone could give a simple justification (i.e. one that doesn't use anything but definition of the Euclidean norm and the basic algebraic properties of the dot product) for assuming WLOG that $\displaystyle x = 0$ and $\displaystyle y = (y_1, 0, ..., 0)$, this would certainly do the trick. Any other permissible solution would be fine too.

Thanks.