Some theorems on convex hypersurfaces in an affine space
The author develops Minkowski-type integral formulae for closed convex hypersurfaces in equiaffine differential geometry. He deduces several characterizations of affine spheres and uniqueness theorems for pairs of hypersurfaces. Unfortunately, it appears that he was not aware of much of the existing relevant literature. For instance, the work of \textit{U. Simon} [Thesis, FU Berlin 1965; Math. Ann. 173, 307-321 (1967; Zbl 0153.230); ibid. 175, 81-88 (1968; Zbl 0153.231)] containing similar investigations is not quoted. In particular, Theorems 4,5,6 of the present paper can be found in Simon's papers. Some of the results appear already in \textit{W. Blaschke}'s work [Vorlesungen über Differentialgeometrie. II. Affine Differentialgeometrie (Berlin 1923)]; see also \textit{W. Blaschke} [Gesammelte Werke Band 4 (Thales 1985; Zbl 0656.53002) and the comments given there)], at least three theorems of the present paper, dealing with pairs of convex hypersurfaces, are new, but they contain assumptions which seem to be artificial (at least, unexplained) from a geometric point of view.
scientific article; zbMATH DE number 4082415