An ellipse is usually defined as the locus of points so that sum of the distances to the two foci is constant. But what are curves called which are defined as the locus of points so that the sum of the distances to three foci is constant? Trilipse? And what about $n$ foci? $n$-lipse? How do these curves look like? Is there any literature about them?
These are called $k$-ellipses.
Yes, there is a literature.
Here is one 2007 reference, which can
lead you to others: "Semidefinite Representation of the $k$-Ellipse,"