Affine geometry and the form of the equation of a hypersurface (Q1821359)

For a hypersurface M of \(R^{n+1}\), represented in the classical form as the zero-set of a function \(F: R^{n+1}\to R\), i.e. \(M=\{X\in R^{n+1}: F(X)=0\},\) the author proposes the question whether there is a ''canonical'' form for the equation of M, \(F(X)=0\) (such an F is called an M-function if the regularity condition \(DF_ p\neq 0\), at each point p of M, also holds). Although the question is not answered in this paper, the author discusses a class of ''preferred forms'' for the equation, in the case where M is non-degenerate in the sense of unimodular affine geometry. In fact, the prescription of a volume element in \(R^{n+1}\), allows to set the problem in terms of well-known geometrical objects from that geometry. However, in many aspects the theory is developed taking as ambient space a real, infinite dimensional Banach space. In this sense, the main result is an extension of a theorem by Berwald and Pick. It asserts: ''Let M be a connected, nondegenerate hypersurface in the Banach space V, with dim(V)\(\geq 3\). If there exists an M-function F such that the Fubini-Pick form \(III^ F\equiv 0\), then M is a quadratic hypersurface.''