  An affine hypersurface (AH) structure is a pair comprising a conformal
structure and a projective structure such that for any torsion-free connection
representing the projective structure the completely trace-free part of the
covariant derivative of any metric representing the conformal structure is
completely symmetric. AH structures simultaneously generalize Weyl structures
and abstract the geometric structure determined on a non-degenerate co-oriented
hypersurface in flat affine space by its second fundamental form together with
either the projective structure induced by the affine normal or that induced by
the conormal Gauss map. There are proposed notions of Einstein equations for AH
structures which for Weyl structures specialize to the usual Einstein Weyl
equations and such that the AH structure induced on a non-degenerate
co-oriented affine hypersurface is Einstein if and only if the hypersurface is
an affine hypersphere. It is shown that a convex flat projective structure
admits a metric with which it generates an Einstein AH structure, and examples
are constructed on mean curvature zero Lagrangian submanifolds of certain
para-K\"ahler manifolds. The rough classification of Riemannian Einstein Weyl
structures by properties of the scalar curvature is extended to this setting.
Known estimates on the growth of the cubic form of an affine hypersphere are
partly generalized. The Riemannian Einstein equations are reformulated in terms
of a given background metric as an algebraically constrained elliptic system
for a cubic tensor. From certain commutative nonassociative algebras there are
constructed examples of exact Riemannian signature Einstein AH structures with
self-conjugate curvature but which are not Weyl and are neither projectively
nor conjugate projectively flat.
