  We develop a differential theory for the polarity transform parallel to that
for the Legendre transform, which is applicable when the functions studied are
"geometric convex", namely convex, non-negative and vanish at the origin. This
analysis may be used to solve a family of first order equations reminiscent of
Hamilton--Jacobi and conservation law equations, as well as some second order
Monge-Ampere type equations. A special case of the latter, that we refer to as
the homogeneous polar Monge--Ampere equation, gives rise to a canonical method
of interpolating between convex functions.
