On surface of Apollonius of two ellipsoids

Apollonius defined the circle as the set of points that have a given ratio μ μ of distances from two given points, where the ratio is not equal to one. In a more general sense, consider two 0-symmetric, bounded, convex bodies K and $$K'$$ K ′ , which define two norms. Their unit balls are K and $$K'$$ K ′ . The surface of Apollonius is defined as the set of points equidistant from the centres of bodies K and $$K'$$ K ′ with respect to the aforementioned norms. In this paper we demonstrate that the surface of Apollonius of two ellipsoids is a quadratic surface. We also examine the circumstances under which this surface becomes a sphere.