  A self-focal point of a Riemannian manifold $(M,g)$ is a point $p$ so that
every geodesic starting from $p$ returns to $p$ at some positive time. It is
called a pole if all geodesics through $p$ are closed, and a non-polar
self-focal point if all geodesics loop back but not all are smoothly closed.
Umbilic points of two dimensional tri-axial ellipsoids are non-polar self-focal
points. Little is known about existence of self-focal points for Riemannian
manifolds of dimension $\geq 3$. We prove that ellipsoids of dimension $\geq 3$
with at least 4 distinct axes have no self-focal points. Certain ellipsoids of
dimension $\geq 3$ with three distinct axes do have non-polar self-focal
points. Ellipsoids with $\leq 2$ distinct axes always have self-focal points.
  Self-focal points play an important role in the study of $L^{\infty}$ norms
of Laplace eigenfunctions. Our results imply that Laplace eigenfunctions on
ellipsoids of dimension $\geq 3$ with at least $4$ distinct axes never achieve
maximal sup-norm growth.
