n - ellipses

Last updated: July 13, 2023
Tags: Geometry, Convexity

[7/12/23 ]  Formulating the problem of finding the largest convex subset (in a way that can be computed) is difficult in general, and I think that limiting the subsets to a certain shape could make things easier.

\(n\)-ellipses, ellipses with \(n\) focal points, are always convex, and versatile as \(n\) varies.

Nested n-ellipses

The resources I intend to use are:

  1. Semidefinite Representations of The K-Ellipse – Nie, Parillo, and Sturmfels
  2. The Geometry of Semidefinite Programming – Sturmfels
  3. On the Approximation of Convex, Closed Plane Curves by Multifocal Ellipses – Erdös, Vincze
  4. n-Ellipses and the Minimum Distance Sum Problem – Sekino

The item by Erdös is exactly what I’m looking for and it suggests that \(n\)-ellipses might be able to approximate any convex, closed plane curve.

  1. wikipedia.org/wiki/N-ellipse  ↩︎