The authors introduce the invariant \(\varepsilon _M\) for definite centroaffine hypersurfaces \(M^n \rightarrow {\mathbb R}^{n+1}\) in terms of the normalized scalar curvature of a plane section \(\pi \) in \(T_pM\). They derive the inequality \[ \varepsilon _M \geq -{\textstyle{\frac{n^2(n-2)}{2(n-1)}}}h(T^{\# },T^{\# }) + {\textstyle{\frac{1}{2}}}(n+1)(n-2), \tag{*} \] where \(h\) is a positive definite metric and \(T^{\# }\) the Tschebychev field. The class of hypersurfaces satisfying the equality in (*) is a very wide one. They prove that this class consists a proper affine hyperspheres centered at the origin. Using some distribution \(D\) on such hypersurfaces, the authors obtain classification theorems under the additional assumption that either \(M^n\) has constant scalar curvature or the distribution \(D^{\perp }\) is integrable. Examples related to equiaffine elliptic spheres in \({\mathbb R}^3\), realizing the equality in (*) are also given.
Some results about polarization and asymptotic spectral invariants of second order Laplace type operators are related to the previously mentioned results.