This is a very interesting paper presenting topics from affine differential geometry of locally strongly convex hypersurfaces. (1) A locally strongly convex affine hypersphere with zero scalar curvature R of the metric is uniquely determined. Based on this result it was meanwhile possible to finish the classification of all locally strongly convex affine spheres of affine constant sectional cuvature K. In fact, Li's proof works for \(K=const\geq 0\) with minor modifications, while the case \(K<0\) was very recently proved by L. Vrancken (Leuven; unpublished). (2) The author gives a partial solution to the so-called affine Bernstein problem (every affine complete, affine maximal surface is an elliptic paraboloid), if one omits at least 4 directions of the affine normal. The proof uses function theoretic methods. (3) Furthermore, global results for compact hyperfaces with boundary and constant curvature functions are presented.