Introduction
A convex function can be described in two ways: by its points (the graph), or by its tangent lines (for each slope, where the tangent crosses the axis). The convex conjugate, or Legendre–Fenchel transform, switches from the first description to the second. It is the mathematical engine behind Lagrangian duality and appears in variational inference, f-divergences (as used in some GANs), and the analysis of regularizers.