Global uniqueness for ovaloids in Euclidean and affine differential geometry (Q1204159)

Consider an ovaloid in real affine 3-space with relative normalization, with relative metric \(G\) and connection \(\nabla\) (induced by the relative normal). The ovaloid is uniquely determined (modulo affine equivalences) by the connection and the Riemannian volume of \(G\). Euclidean interpretations of this result give the global uniqueness results of Minkowski, Cohn-Vossen and Grove. In equiaffine geometry one needs only \(\nabla\) to determine the ovaloid. A proof in higher dimension is unknown to the author.