Why is the following scaling good for the general case?

Question

In the following post how come scaling to $x_0=0$ and $r=1$ can help to prove for any $r,x_0$?

Is there a general rule for when is it OK to scale?

Answer 1

You can get away with scaling and translating whenever your problem is invariant under transformations of the form $x\mapsto \alpha x + c$ where $\alpha$ is a real number (or, more generally, an element of the underlying field) and $c$ is a vector. Given how many problems respect such transformations, and how averse some mathematicians are to constants other than $0$ and $1$, this is a pretty common trick. More generally, this is what is happening whenever someone writes "without loss of generality" - they are using a symmetry in the problem to extrapolate a proof from fewer cases.

With your question in particular, consider the map $$f(x)=\frac{1}{r}\cdot (x-x_0)$$ You can check that $f$ has an inverse of a similar form. Then, you have that $f(x_0)$ is the origin and $f[C]$ is a convex set in the same space. Moreover, as the above scales distances by $\frac{1}r$, you get that $d(f(x_0),f[C])=1$. Finally, if a ball of radius $r$ around $x_0$ intersects $C$ in $n$ places, then the ball of radius $1$ around $f(x_0)$ intersects $f[C]$ in $n$ places as well.

At this point, you can just apply the argument that you know: If $x_0$ is the origin and $d(x_0,C)=1$, then $\{y\in X:\|y\|\leq r\}\cap C$ has at most one point. However, since every other situation can be reduced to one like this without changing the size of this intersection, the statement is true in general.