Let x: M → An+1 be a locally strongly convex hypersurface, given as the graph of a locally strongly convex function xn+1 = z(x1, ..., xn ). In this paper we prove a Bernstein property for hypersurfaces which are complete with respect to the metric G # = ∑ (â2 z/âx1 âxj) dxi dxj and which satisfy a certain Monge-Ampere type equation. This generalises in some sense the earlier result of Li and Jia for affine maximal hypersurfaces of dimension n = 2 and n = 3 (Li, A.-M., Jia, F.: A Bernstein property of affine maximal hypersurfaces. Ann. Glob. Anal. Geom. 23, 359-372 (2003)), related results (Li, A.-M., Jia, F.: Locally strongly convex hypersurfaces with constant affine mean curvature. Diff. Geom. Appl. 22(2), 199-214 (2005)) and results for n = 2 of Trudinger and Wang (Trudinger, N.S., Wang, X.-J.: Bernstein-Jorgens theorem for a fourth order partial differential equation. J. Partial Diff. Equ. 15(2), 78-88 (2002)).