  Let $U_c(H)={u: u is unitary and u-1 is compact}$ stand for the unitary
Fredholm group. We prove the following convexity result. Denote by $d_\infty$
the rectifiable distance induced by the Finsler metric given by the operator
norm in $U_c(H)$. If $u_0,u_1,u\in U_c(H)$ and the geodesic $\beta$ joining
$u_0$ and $u_1$ in $U_c(H)$ verifies $d_\infty(u,\beta)<\pi/2$, then the map
$f(s)=d_\infty(u,\beta(s))$ is convex for $s\in[0,1]$. In particular the
convexity radius of the geodesic balls in $U_c(H)$ is $\pi/4$. The same
convexity property holds in the $p$-Schatten unitary groups $U_p(H)={u: u is
unitary and u-1 is in the p-Schatten class}$, for $p$ an even integer, $p\ge 4$
(in this case, the distance is strictly convex). The same results hold in the
unitary group of a $C^*$-algebra with a faithful finite trace. We apply this
convexity result to establish the existence of curves of minimal length with
given initial conditions, in the unitary orbit of an operator, under the action
of the Fredholm group. We characterize self-adjoint operators $A$ such that
this orbit is a submanifold (of the affine space $A+K(H)$, where $K(H)$=compact
operators).
