The two foci of an ellipse $p, q$ are defined so that $d(x, p) + d(x, q) = c$ for some constant $c$ and all $x$ lying on the boundary of an ellipse.
In general, people have studied n-ellipses, where we fix $n$ points $p_i$ and a constant $c$ and consider the solutions of $x$ to the equation $\sum_{i = 1}^n d(x, p_i) = c$. The solutions define a curve in $\mathbb{R}^2$.
Is there a further generalization of this that is studied for higher dimensions? Namely, if I give you $m$ points in $\mathbb{R}^n$ so that the $m$ points form an $n$-dimensional polytope and there is some vertex, say $v$ such that $v$ is connected to every other vertex then we can vary $v$ around to keep the total surface area (sum of $n-1$ dimensional volume of facets) of the polytope constant. What shape in general does the set of all such movements of $v$ define?
