  We study convexity properties of distance functions in Finsler unitary
groups, where the Finsler structure is defined by translation of the
$p$-Schatten norm on the Lie algebra. As a result we prove the existence of
circumcenters for sets with radius less than $\pi/2$ in several metrics. This
result is applied to a fixed point property and to quantitative metric bounds
in certain rigidity problems. Bounds for convexity, existence of circumcenters
and rigidity are shown to be optimal.
