Smooth solutions of the eikonal equation and the behaviour of local minima of the distance function

We study smooth solutions of the eikonal equation. To do this, we investigate the problem of geometric-topological properties of the singularities of the distance function and the regular set. We establish a connection between the caustic and domains where the number of local minima of the distance function is constant. We pose a number of problems about reflecting surfaces bringing light to a single point (a focus) and introduce the notions of generalized ellipsoids and paraboloids.