A classical ellipse has two foci (I hope my English is correct saying this) and admits the Klein group as isometry group, whose elements are the identity, the reflexions along the small and larges axes, and the central symmetry whose fixed point is the middle of the segment the two foci define. Those symmetries preserve the set of foci globally, that is, they induce a permutation thereof.
I would like to know if similarly, given a set of $N+1$ points $F_{i}$ for $i\in[0,N]\cap\mathbb{Z}$ of the $N$-dimensional euclidean space, the isometry group of the locus of the points $M$ such that $\sum_{i=0}^{N}MF_{i}=K>0$ induces a permutation of the points $F_{i}$.
Can we even go farther and consider and uncountable set of points and prove that $C_{\rho}(K):=\{z\in\mathbb{C}\mid\int_{0}^{1}\vert z-\rho e^{2i\pi t}\vert dt=K>0\}$ is a circle for all $\rho\geq 0$?
Here is a counterexample showing (thanks to Desmos) that in $\mathbb{R}^N$ with $N=2$, a super-ellipse defined by $N+1=3$ foci isn't preserved by any symmetry because the foci are in the same situation (sorry, points are in $(\mathbb{Z}/2)^2$...)
But, have I well understood your question ?