On a projectively minimal hypersurface in the unimodular affine space (Q1089929)

\textit{W. Blaschke} [Vorlesungen über Differentialgeometrie. II. Affine Differentialgeometrie (1923)] defined the projective area functional using the Fubini-Pick's cubic form. A projectively minimal surface is a critical point of this functional. In this paper the author derives a differential equation defining a projectively minimal hypersurface in the affine space \(A^{n+1}\). Making use of this he obtains non-trivial examples of projectively minimal hypersurfaces which are closed in \(A^{n+1}\). He also proves the following theorems. Theorem 1. (1) (H. Behnke) Every affine sphere in \(A^ 3\) is projectively minimal. (2) If the surface is compact, strongly convex and projectively minimal, then it is a quadric. Theorem 2. Assume the surface to be strongly convex and closed in \(A^ 3\). If it is both affinely and projectively minimal, then it is a paraboloid.