Apollonius defined the circle as the set of points that have a given ratio $\mu$ of distances from two given points, where $\mu\not=1$. Generally, consider two $0$-symmetric, bounded, convex bodies $K$ and $K'$, which define two norms, whose unit balls are $K$ and $K'$. The surface of Apollonius is the set of points equidistant from centers of bodies $K$ and $K'$ with respect to these norms. We prove that the surface of Apollonius of two ellipsoids is a quadratic surface and we examine when this surface becomes a sphere.
Mathematics Subject Classification (2010). Primary 51N20, 52A20; Secondary 51N10, 52A10.