Let $U_c\left(\right)$ = u: u unitary and u-1 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 $U_c\left(\right)$. If $u₀,u₁,u\in U_c\left(\right)$ and the geodesic β joining u₀ and u₁ in $U_c\left(\right)$ satisfy $d_{\infty }(u,\beta )<\pi /2$, then the map $f\left(s\right)=d_{\infty }(u,\beta \left(s\right))$ is convex for s ∈ [0,1]. In particular, the convexity radius of the geodesic balls in $U_c\left(\right)$ is π/4. The same convexity property holds in the p-Schatten unitary groups $U_p\left(\right)$ = u: u unitary and u-1 in the p-Schatten class...