Hypersurfaces of constant Gauss–Kronecker curvature with Li-normalization in affine space

For convex hypersurfaces in the affine space An+1 (n≥ 2), A.-M. Li introduced the notion of α-normal field as a generalization of the affine normal field. By studying a Monge–Ampère equation with gradient blowup boundary condition, we show that regular domains in An+1, defined with respect to a proper convex cone and satisfying some regularity assumption if n≥ 3 , are foliated by complete convex hypersurfaces with constant Gauss–Kronecker curvature relative to the Li-normalization. When n= 2 , a key feature is that no regularity assumption is required, and the result extends our recent work about the α= 1 case.