  In this article we study convexity properties of distance functions in
infinite dimensional Finsler unitary groups, such as the full unitary group,
the unitary Schatten perturbations of the identity and unitary groups of finite
von Neumann algebras. The Finsler structures are defined by translation of
different norms on the tangent space at the identity. We first prove a
convexity result for the metric derived from the operator norm on the full
unitary group. We also prove strong convexity results for the squared metrics
in Hilbert-Schmidt unitary groups and unitary groups of finite von Neumann
algebras, in both cases the tangent spaces are endowed with an inner product
defined with a trace. These results are applied to fixed point properties and
to quantitative metric bounds in certain rigidity problems. Radius bounds for
all convexity and fixed point results are shown to be optimal.
