On the ∞-Norm of the Cubic form of Complete Hyperbolic Affine Hyperspheres

Abstract

Let \({M_n \subset \mathbb R^{n+1}}\) be a complete hyperbolic affine hypersphere with mean curvature H, \({H < 0}\), and let C be its cubic form. We derive a differential inequality and an upper bound on the scalar function \({||C||_{\infty}}\) defined by the fiber-wise maximum of the value of C on the unit sphere bundle of M. The bounds are attained for the affine hyperspheres which are asymptotic to a simplicial cone. The results have applications in conic optimization.