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

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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.

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  1. LJK, Université Grenoble 1/CNRS, 51 rue des Mathématiques, BP53, 38041, Grenoble cedex 09, France

    Roland Hildebrand

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  1. Roland Hildebrand
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Correspondence to Roland Hildebrand.

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Hildebrand, R. On the ∞-Norm of the Cubic form of Complete Hyperbolic Affine Hyperspheres. Results. Math. 64, 113–119 (2013). https://doi.org/10.1007/s00025-012-0301-7

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  • DOI: https://doi.org/10.1007/s00025-012-0301-7

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