Given two probability measures $\mu$ and $\nu$ we consider a mass
transportation mapping $T$ satisfying 1) $T$ sends $\mu$ to $\nu$, 2) $T$ has
the form $T = \varphi \frac{\nabla \varphi}{|\nabla \varphi|}$, where $\varphi$
is a function with convex sublevel sets. We prove a change of variables formula
for $T$. We also establish some a priori estimates for $T$, and a new form of
the parabolic maximum principle. In addition, we discuss relations to the
Monge--Kantorovich problem, curvature flows theory, and parabolic nonlinear
PDE's.