Is it true that $k=n-1$ and $\{f(d_i)\}=\{n-1,\ldots,1\}$ if and only if $A$ is a set of equidistant points on a line or a circle?
Is it true that $k=n-1$ and $\{f(d_i)\}=\{n-1,\ldots,1\}$ if and only if $A$ is a set of equidistant points on a line or a circle?
This was proved by Clemen, Dumitrescu, and Liu [CDL25], who observed that equidistant points on a short circular arc on a circle of radius $1$, together with the centre, are also an example.