Relative Tchebychev hypersurfaces which are also translation hypersurfaces

Abstract

The class of relative Tchebychev hypersurfaces extends the class of affine hyperspheres in Blaschke geometry. This paper aims at finding examples for relative Tchebychev hypersurfaces among the translation hypersurfaces. A complete local classification is given. Specializing from relative to centroaffine geometry, the list reduces to paraboloids.