Affine hypersurfaces with self congruent center map

In this paper, we study locally strictly convex affine hypersurfaces for which the center map is centroaffine congruent with the original hypersurface. By the equiaffine support function ρ  , we show that the hypersurface is locally isometric to a warped product R×|ρ|N, where the gradient direction of ρ   is along RR. As a main result, we complete the classification when gradρ is the eigenvector of affine shape operator, which shows how to explicitly construct such hypersurfaces starting from one (or two) low dimensional affine hypersphere(s).