Two results in the affine hypersurface theory
The first result gives a local classification of locally strongly convex affine spheres in affine 3-spaces with equiaffine constant curvature metric. This result was independently proved by \textit{A.-M. Li} and \textit{G. Penn} [Result. Math. 13, No.3/4, 308-317 (1988; Zbl 0646.53009)] and partly generalized to higher dimensions by \textit{A.-M. Li} [Acta Math. Sin., New Ser. 5, 345-354 (1989)] and \textit{J. H. Yu} [Affine hyperspheres with constant sectional curvature in \(A^ 4\) (Preprint Sichuan University, China)] for locally strongly convex hypersurfaces. Meanwhile all affine 2-spheres (also with indefinite metric) are classified [the reviewer, `Local classification of twodimensional affine spheres with constant curvature metric, Differ. Geom. Appl. (Brno) (to appear)], which is based on results of \textit{M. Magid} and \textit{P. Ryan} [Flat affine spheres in \({\mathbb{R}}^ 3\) (to appear)]. The second result is the affine version of the global existence and uniqueness of Minkowski for ovaloids. This affine-Euclidean relation was already known to Blaschke [\textit{W. Blaschke}, Gesammelte Werke. Vol. 4 (Essen 1985; Zbl 0656.53002), see the commentary there, p. 53 and p. 63 for further references]. A generalization of the uniqueness part can be found in [the reviewer, Recent developments in affine differential geometry, differential geometry and its applications, Proc. Conf., Dubrovnik/Yugosl. 1988, 327-347 (1989)].
scientific article; zbMATH DE number 4120920