  The Moreau envelope is one of the key convexity-preserving functional
operations in convex analysis, and it is central to the development and
analysis of many approaches for convex optimization. This paper develops the
theory for an analogous convolution operation, called the polar envelope,
specialized to gauge functions. Many important properties of the Moreau
envelope and the proximal map are mirrored by the polar envelope and its
corresponding proximal map. These properties include smoothness of the envelope
function, uniqueness and continuity of the proximal map, which play important
roles in duality and in the construction of algorithms for gauge optimization.
A suite of tools with which to build algorithms for this family of optimization
problems is thus established.
