  In this article, we study a convex embedding for the Euler problem of two
fixed centers for energies below the critical energy level. We prove that the
doubly-covered elliptic coordinates provide a 2-to-1 symplectic embedding such
that the image of the bounded component near the lighter primary of the
regularized Euler problem is convex for any energy below the critical Jacobi
energy. This holds true if the two primaries have the equal mass, but does not
holds near the heavier body.
