The centroaffine volume of generalized geodesic balls under inversion at the sphere (Q1411452)

The Taylor expansion for the volume of geodesic balls under the exponential mapping on analytic Riemannian manifolds \((M,g)\) is well known. In [Results Math. 43, 205-234 (2003; Zbl 1057.53021)], \textit{N. Bokan}, \textit{M. Djoric} and \textit{U. Simon} investigated a more general structure \((M,D,g)\), where \(D\) is a torsion-free and Ricci-symmetric connection, and calculated the Taylor expansion up to order \((n+4)\) for the volume in case that all metric notions are Riemannian, while the exponential mapping is induced from the connection \(D\). Of course for the structure \((M,D,g)\) the coefficients of the Taylor expansion are much more complicated than in the Riemannian case. Now in this recent and excellently written paper the author studies centro affine hypersurfaces in Euclidean space, their geometric invariants which appear in the rather complicated coefficient of order \((n+4)\), and their behaviour under inversion at the unit sphere. The obtained nice results complement applications in the above cited paper.