Abstract. A closed convex surface S in IE d is an ellipsoid if and only if for any x, y ∈ S there is an affinity mapping x onto y and a neighbourhood of x in S onto a neighbourhood of y in S. Keywords. Characterization of ellipsoids, ellipsoids, Helmholtz–Lie type results, local affine transformation, functional equations, floating bodies, affine differential geometry, affine distance, Santaló point, Gauss curvature, local supporting properties, Baire categories. AMS-Classification 1991. 52A10, 52A20, 53A05, 53A1