The convex analysis of unitarily invariant matrix functions

A fundamental result of von Neumann's identies unitarily invariant matrix norms as symmetric gauge functions of the singular values. Identifying the subdierential of such a norm is important in matrix approximation algorithms, and in studying the geometry of the corresponding unit ball. We show how to reduce many convex-analytic questions of this kind to questions about the underlying gauge function, via an elegant Fenchel conjugacy formula. This approach also allows such results to be extended to more general unitarily invariant matrix functions