A Remarkable Measure Preserving Diffeomorphism between two Convex Bodies in <Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let K and T be convex open bounded bodies in R . We will discuss in this note C1-diffeomorphic bijections8:K → T with constant determinant of the Jacobian. So, if volK = vol T our maps are measure preserving. There are several such distinguished maps. For example, a Knothe map ([K]) 8:K → T has a triangular Jacobian with a positive diagonal and may be naturally associated with any fixed orthogonal basis in R . So, there are many different such maps (see [MSch, App. 1], or [Bo] where a Knothe map was used to estimate integrals of polynomials over convex bodies). Recently, the so-called Brenier map [Br] was introduced and used successfully. We call b:K → T the Brenier map if there is a convex function f ∈ C2(K) defined on K such that ∇f = b:K → T has the above properties (bijection which preserves ratio of volumes). Existence and uniqueness of such a map was proved in [Br] (see another proof in [McC1] where more general facts were also proved, and the proof of regularity in [C1-3]). Naturally, the Jacobian matrix Jac b = Hess f is a symmetric positive definite matrix. One of the (many) applications of such maps is in proving the Brunn– Minkowski inequality (and also in analyzing equality cases and their stability).