A Remarkable Measure Preserving Diffeomorphism between two Convex Bodies in ℝⁿ

We prove that for any two convex open bounded bodies K and T there exists a diffeomorphism f : K → T preserving volume ratio (i.e. with constant determinant of the Jacobian) and such that the Minkowski sum K + T { x + f (x) | x ∈ K }. As an application of this method, we prove some of the Alexandov–Fenchel inequalities.