Generalized Legendre transform of conformally flat metrics

In the calculus of variations, an important role is played by the Minkowski duality, or the Legendre transform of convex functions. We consider weakly regular, conformally flat Riemannian metrics of nonnegative curvature defined on the $n$-dimensional unit sphere. For this class of metrics, an analog of the Legendre transformation is introduced and studied in detail.