A further characterization of ellipsoids (Q817154)

The authors consider a hyperovaloid in unimodular affine space \(\mathbb{A}^{n+1}\). Denote by \(\lambda_1, \dots, \lambda_n\) the affine principal curvatures and by \(L_1,\dots, L_n\) their normed elementary symmetric functions. The authors improve a result from the monograph [\textit{A. M. Li, U. Simon} and \textit{G. Zhao}, Global Affine Differential Geometry of Hypersurfaces. De Gruyter Expositions Math. 11 (1993; Zbl 0808.53002)]; they consider linear relations of the type \[ L_r = \sum_{j=0}^{r-1} C_j L_j \] with real positive constants \(C_j \) to characterize ellipsoids. The proof uses affine Minkowski integral formulas and inequalities for the curvature functions \(L_r\).