The stability of the unit balls of symmetric and unitarily invariant norms Original Research Article

A compact convex set K is called stable if the midpoint mapping, K × K → K, (x, y) → (x + y)/2, is open. The main result asserts that the stability of the closed unit ball of a unitarily invariant norm is equivalent to the stability of the closed unit ball of the associated symmetric gauge function. This result, as well as other pointwise related results, are obtained using a recently found close relationship between the facial structures of those two kinds of unit balls.