We develop a differential theory for the polarity transform parallel to that of the Legendre transform, which is applicable when the functions studied are “geometric convex”, namely, convex, non-negative, and vanish at the origin. This analysis establishes basic tools for dealing with this duality transform, such as the polar subdifferential map, and variational formulas. Another crucial step is identifying a new, non-trivial, sub-class of C2 functions preserved under this transform. This analysis leads to a new method for solving many new first order equations reminiscent of Hamilton–Jacobi and conservation law equations, as well as some second order equations of Monge–Ampère type. This article develops the theory of strong solutions for these equations which, due to the nonlinear nature of the polarity transform, is considerably more delicate than its counterparts involving the Legendre transform. As one application, we introduce a polar form of the homogeneous Monge–Ampère equation that gives a dynamical meaning to a new method of interpolating between convex functions and bodies. A number of other applications, e.g., to optimal transport and affine differential geometry are considered in sequels.
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