The gradient map associated to a regular open convex cone gives a diffeomorphism from the cone onto its dual cone. If the cone is homogeneous, the inverse of the map is known to be equal to the gradient map associated to the dual cone. However, we show that this is no longer true for a general case by presenting a simple counterexample.
Hideyuki Ishi. "The gradient maps associated to certain non-homogeneous cones." Proc. Japan Acad. Ser. A Math. Sci. 81 (3) 44 - 46, March 2005. https://doi.org/10.3792/pjaa.81.44