  Friedlander, Mac\^{e}do, and Pong recently introduced the projected polar
proximal point algorithm (P4A) for solving optimization problems by using the
closed perspective transforms of convex objectives. We analyse a generalization
(GP4A) which replaces the closed perspective transform with a more general
closed gauge. We decompose GP4A into the iterative application of two separate
operators, and analyse it as a splitting method. By showing that GP4A and its
under-relaxations exhibit global convergence whenever a fixed point exists, we
obtain convergence guarantees for P4A by letting the gauge specify to the
closed perspective transform for a convex function. We then provide
easy-to-verify sufficient conditions for the existence of fixed points for the
GP4A, using the Minkowski function representation of the gauge. Conveniently,
the approach reveals that global minimizers of the objective function for P4A
form an exposed face of the dilated fundamental set of the closed perspective
transform.
